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\begin{document}

\begin{abstract}
We show that for any lattice Veech group in the mapping class group $\Mod(S)$ of a closed surface $S$, the associated $\pi_1S$--extension group is a hierarchically hyperbolic space.
\end{abstract}

\maketitle

\section{Introduction}

Let $S$ be a closed, connected, oriented surface of genus at least $2$.  For any group $G$, a $\pi_1S$--extension of $G$ is a group $\Gamma$ fitting into a short exact sequence:
\[ 1 \to \pi_1S \to \Gamma \to G \to 1.\]
Such groups are precisely the fundamental groups of $S$--bundles.
Each such extension induces, and is in fact determined up to isomorphism by, an associated monodromy homomorphism $G\to \Mod^{\pm}(S)\cong\mathrm{Out}(\pi_1S)$ to the extended mapping class group of $S$.
An important, intriguing problem is to decide how geometric properties of $\Gamma$ (or the bundle) are reflected by properties of the monodromy.

Thurston's geometrization (see \cite{Otal-ThurstonHyp}) provides a very strong connection,  proving that when $G \cong \mathbb Z$ the associated surface bundle over the circle admits a hyperbolic structure if and only if the image of the monodromy is generated by a pseudo-Anosov mapping class.  A generalization of this to the world of coarse geometry states that
a $\pi_1S$--extension of any group $G$ is {\em word-hyperbolic} if and only if the monodromy has finite kernel and its image in $\Mod(S)$ is {\em convex cocompact}  in the sense of Farb and Mosher (see \cite{FM:CC,hamenstadt:WHSG,MjSardar}).

In the classical setting of Kleinian groups, convex cocompactness is a special case of the much more general notion of geometric finiteness.
However, there is currently no precise analogue of geometric finiteness in the context of mapping class groups; nor is there a proposal for how such a notion should relate, as in the case of convex cocompactness, to the (coarse) geometry of associated extension groups.
Nevertheless, it is evident that a {\em lattice Veech group} $G< \Mod(S)$ should qualify as geometrically finite with respect to {\em any} definition (c.f.~Mosher \cite[\S6]{Mosher:Problems}).
This paper explores this vein by providing the first analysis of the coarse geometry of these extension groups. Our main theorem is the following.

\begin{theorem} \label{T:main simple} For any lattice Veech group $G < \Mod(S)$, the $\pi_1S$--extension group $\Gamma$ of $G$ admits the structure of a hierarchically hyperbolic space.
\end{theorem}

The hope is that any reasonable definition of geometric finiteness in mapping class group should have some ``hyperbolic'' implications for surface group extensions. Theorem~\ref{T:main simple} provides precisely this sort of result in the prototypical case of lattice Veech groups and thus hints at a possible theory of geometric finiteness in general.
This possibility is expounded upon in \S\ref{section:musings-on-geom-finte} below.

The notion of hierarchical hyperbolicity was defined by Behrstock, Hagen, and Sisto \cite{BHS:HHS1} and motivated by the seminal work of Masur and Minsky \cite{MM:CC2}.
In short, it provides a framework and toolkit for understanding the coarse geometry of a space/group in terms of interrelated  hyperbolic pieces.
More precisely, a {\em hierarchically hyperbolic space (HHS)} structure on a metric space $X$ is a collection of hyperbolic spaces $\{\C(W)\}_{W \in \mathfrak S}$, together with additional data, that is meant to capture the coarse geometry of $X$. The only piece of data that we need for the discussion below is that there is a \emph{maximal} hyperbolic space (technically, $\nest$--maximal).
See \S\ref{S:combinatorial HHS} for details or \cite{BHS:HHS1,BHS:HHS2,HHS_survey} for more examples and discussion.

Showing that a space/group is a hierarchically hyperbolic space gives access to several results regarding, for example, quadratic isoperimetric function \cite{Bow:rigidity_MCG}, asymptotic dimension \cite{BHS:asdim}, stable and quasiconvex subsets and subgroups \cite{ABD:stable, RST18}, quasiflats \cite{BHS:quasiflats}, bordifications and automorphisms \cite{DHS}, and quasi-isometric embeddings of nilpotent groups \cite{BHS:HHS1}.

In order to describe the HHS structure constructed in Theorem~\ref{T:main simple} and discuss its  consequences, we must first recall some of the structure of Veech groups and their extensions.

\subsection{Extensions of Veech groups}

Let $G < \Mod(S)$ be a lattice Veech group and $\Gamma = \Gamma_G$ the associated extension group. The HHS structure that we will build on $\Gamma$ is motivated by certain natural subgroups. First note that (up to finite index) $\Gamma$ is naturally the fundamental group of an $S$--bundle $\bar E/\Gamma$ over a compact surface with boundary (see \S\ref{sec:bundles-total-space} for details and notation).  Each boundary component of $\bar E / \Gamma$ is the mapping torus of a multi-twist on $S$, and is thus a {\em graph manifold}:  the tori in the JSJ decomposition are suspensions of the multi-twist curves.  Graph manifolds admit HHS structures \cite{BHS:HHS2} where the maximal hyperbolic space is the Bass--Serre tree dual to the JSJ decomposition, and all other hyperbolic spaces are either quasi-lines or quasi-trees (obtained by coning off the universal covers of the base orbifolds of the Seifert pieces).
The stabilizers of the vertices of the Bass--Serre tree are called \emph{vertex subgroups}, and are precisely the fundamental groups of the Seifert pieces of the JSJ decomposition.

We summarize the main results of this paper (Theorem~\ref{T:hyperbolicity of hat E}, Corollary \ref{cor:Cayley_graph}, Proposition \ref{prop:loxodromics}, Theorem \ref{thm:E_is_chhs}, Lemmas \ref{L: empty simplex case}--\ref{L:remaining simplices}) in the following theorem.  To state it, we let $\vtx$ be the disjoint union of the vertices of all Bass--Serre trees associated to the boundary components of the universal cover $\bar E$ of the $S$--bundle above.

\begin{theorem} \label{T:main full} Suppose $G<\Mod(S)$ is a lattice Veech group with extension group $\Gamma$ and let $\Upsilon_1,\ldots,\Upsilon_k < \Gamma$ be representatives of the conjugacy classes of vertex subgroups.
Then $\Gamma$ admits an HHS structure for which the maximal hyperbolic space $\hat E$ is quasi-isometric to the Cayley graph of $\Gamma$ coned off along the cosets of $\Upsilon_1,\ldots,\Upsilon_k$.  Each non-maximal hyperbolic space is a quasi-tree (including quasi-lines and bounded spaces). Furthermore, $\Gamma$ acts isometrically on $\hat E$ and every element not conjugate into one of $\Upsilon_1,\ldots,\Upsilon_k$ acts loxodromically, i.e.~with positive (asymptotic) translation length.
\end{theorem}

The space $\hat E$ actually contains a copy of each Bass--Serre tree from the peripheral graph manifolds.  However, these are all distorted inside of $\hat E$.

\begin{remark} While $\Gamma$ acts on $\hat E$ and the vertex groups come with associated quasi-trees that they act on, the vertex groups also come with certain quasi-lines that they do {\em not} act on. (This is similar to the case of graph manifolds.)
Thus the theorem does not make $\Gamma$ into a {\em hierarchically hyperbolic {\bf group}}.
\end{remark}

There are several consequences of Theorem~\ref{T:main full} and the more detailed description of the HHS structure arising from Theorem \ref{thm:E_is_chhs}; we highlight two of them here.
First, hierarchically hyperbolic spaces are coarse median \cite{Bow:rigidity_MCG}, and therefore have quadratic isoperimetric function \cite{Bow:coarse_median}, so that:

\begin{corollary}
The extension group $\Gamma$ has a quadratic Dehn function.
\end{corollary}

Moreover, one can readily check that the maximal number of pairwise orthogonal domains having unbounded associated hyperbolic space is 2 (this uses the description of orthogonality from Definition \ref{defn:orth}). Therefore, in view of \cite{BHS:HHS1} and \cite{BHS:quasiflats}, we get the following:

\begin{corollary}
Each top-dimensional quasi-flats in $\Gamma$ has dimension $2$ and is contained in a finite-radius neighborhood of finitely many cosets of vertex subgroups.
\end{corollary}

We note that the corollary could be stated more precisely in terms of \emph{standard orthants}, and we remark that the analogous statement for graph manifolds is due to Kapovich--Leeb \cite{KL:qi_decomposition}.

\subsection{Outline and proofs}
Let us briefly outline the paper and comment on the main structure of the proofs.
See the beginning of each section for more discussion.
In \S\ref{S:Definitions} we review necessary background material. Then in \S\ref{S:setup} we introduce all the objects and notation that will be used in the rest of the paper.  In particular, we define the space $\bar E$, which is a quasi-isometric model for the Veech group extension $\Gamma$. Here we also introduce the space $\hat E$, roughly a coned-off version of $\bar E$, that will ultimately serve as the maximal hyperbolic space for the HHS structure of $\Gamma$.

In \S\ref{S:slim triangles} we prove the space $\hat E$ is hyperbolic, a major part of the proof of Theorem~\ref{T:main full}, and classify the elements of $\Gamma$ acting loxodromically.
Roughly, $\hat E$ is something like a bundle over a coned-off Teichm\"uller disk, where all fibers are objects related to flat geometry on surfaces. More precisely, each fiber is either the universal cover of $S$ with a flat metric, or a tree (in fact the Bass--Serre tree mentioned above) dual to a certain foliation.  To prove hyperbolicity, we exploit features of flat geometry to construct explicit paths that form thin triangles, and then use the ``Guessing geodesics'' criterion (Proposition~\ref{T:bowditch}) due to Masur--Schleimer \cite{MS:disks} and Bowditch \cite{Bowditch:Hyp}. The proofs involve a careful analysis of various types of triangles in the universal cover of a flat surface (which can be more complicated than one might expect).

In \S\S\ref{S:projections}--\ref{S:combinatorial HHS} we prove that the extension group $\Gamma$ is hierarchically hyperbolic by utilizing a combinatorial criterion from \cite{comb_HHS}. Besides hyperbolicity of $\hat E$, the other hard part of the criterion is an analogue of Bowditch's fineness condition from the context of relative hyperbolicity. Its geometric interpretation is roughly that two cosets of vertex subgroups as above have bounded coarse intersection, aside from the ``obvious'' exception when the cosets correspond to vertices of the same Bass--Serre tree within distance 2 of each other. To this end, in \S\ref{S:projections} we construct and study certain ``projection'' maps  and prove that various pairs of subspaces of $\bar E$ have bounded projection onto each other. Here we also construct the hyperbolic spaces that will comprise the rest of the HHS structure on $\Gamma$.
With these geometric tools in hand, we have all the input needed to apply the criterion from \cite{comb_HHS}. This is carried out in \S\ref{S:combinatorial HHS}, where the proofs mostly consist of combinatorial considerations, and this will conclude the proof that $\Gamma$ is an HHS.

\subsection{Motivation: Geometric finiteness and convex cocompactness}
\label{section:musings-on-geom-finte}

We conclude the introduction with some speculative discussion.
For Kleinian groups---that is, discrete groups of isometries of hyperbolic $3$--space---the notion of geometric finiteness is important in the deformation theory of hyperbolic $3$--manifolds by the work of Ahlfors \cite{Ahlfors} and Greenberg \cite{Greenberg}.
While the definition has many formulations (see \cite{Marden,Maskit:bdy,thurston:GT,Bow:GF}), roughly speaking a group is geometrically finite if it acts cocompactly on a convex subset of hyperbolic $3$--space minus horoballs invariant by parabolic subgroups.  When there are no parabolic subgroups, geometric finiteness reduces to convex cocompactness: a cocompact action on a convex subset of hyperbolic $3$--space.

There is no deformation theory for subgroups of mapping class groups.
However, Farb and Mosher \cite{FM:CC} defined a notion of convex cocompactness for $G < \Mod(S)$ in terms of the action on Teichm\"uller space $\Teich(S)$.  Their definition requires that $G$ acts cocompactly on a quasi-convex subset for the Teichm\"uller metric, while Kent and Leininger later proved a variety of equivalent formulations analogous to the Kleinian setting \cite{KL:survey,KL:CC,KL:uniform}. Farb and Mosher proved that convex cocompactness is equivalent to hyperbolicity of the associated extension group $\Gamma_G$ (with monodromy given by inclusion) when $G$ is virtually free. This equivalence was later proven in general by Hamenst\"adt \cite{hamenstadt:WHSG} (see also Mj--Sardar \cite{MjSardar}), though at the moment the only known examples are virtually free.
Convex cocompact subgroups are also precisely the stable subgroups of $\Mod(S)$ \cite{DT15}.

The coarse nature of Farb and Mosher's formulation reflects the fact that the Teichm\"uller metric is far less well-behaved than that of hyperbolic $3$--space. Quasi-convexity in the definition is meant to help with the lack of nice {\em local} behavior of the Teichm\"uller metric. It also helps with the {\em global} lack of Gromov hyperbolicity (see Masur--Wolf \cite{masur:NH}), as cocompactness of the action ensures that the quasi-convex subset in the definition is Gromov hyperbolic (see Kent--Leininger \cite{KL:CC}, Minsky \cite{Minsky:quasiproj}, and Rafi \cite{rafi:HT}).

The inclusion of reducible/parabolic mapping classes in a subgroup $G< \Mod(S)$ brings the {\em thin parts} of $\Teich(S)$ into consideration; these subspaces contain higher rank quasi-flats and even exhibit aspects of positive curvature (see Minsky~\cite{Minsky:product}).  This is a main reason why extending the notion of convex cocompactness to geometric finiteness is complicated. These complications are somewhat mitigated in the case of lattice Veech groups. Such subgroups are stabilizers of isometrically and totally geodesically embedded hyperbolic planes, called {\em Teichm\"uller disks}, that have finite area quotients.  Thus, the intrinsic hyperbolic geometry agrees with the extrinsic Teichm\"uller geometry, and as a group of isometries of the hyperbolic plane, a lattice Veech group {\em is} geometrically finite.  This is why these subgroups serve as a test case for geometric finiteness in the mapping class group.

The action of $\Mod(S)$ on the {\em curve graph}, which is Gromov hyperbolic by work of Masur--Minsky \cite{MM:CC1}, provides an additional model for these considerations. Specifically, convex cocompactness is equivalent to the orbit map to the curve graph $\C(S)$ being a quasi-isometric embedding with respect to the word metric from a finite generating set (see Kent--Leininger \cite{KL:CC} and Hamenst\"adt \cite{hamenstadt:WHSG}).

Viewing geometric finiteness as a kind of ``relative convex cocompactness'' for Kleinian groups suggests an interesting connection with the the curve complex formulation of convex cocompactness.
This connection is illustrated by the following theorem of Tang \cite{tang2019affine}.  If $G < \Mod(S)$ is a lattice Veech group stabilizing a Teichm\"uller disk $D \subset \Teich(S)$, then there is a $G$--invariant choice of horoballs in $D$ which are precisely where $D$ ventures into the thin parts of $\Teich(S)$.  We write $D^{el}$ for the path metric space obtained by coning off all horoballs to have diameter $1$.

\begin{theorem}[Tang] \label{T:Tang} For any lattice Veech group $G < \Mod(S)$ stabilizing a Teichm\"uller disk $D \subset \Teich(S)$, there is a $G$--equivariant quasi-isometric embedding $D^{el} \to \C(S)$.
\end{theorem}

Farb \cite{Farb:rel} showed that non-cocompact lattices in the group of isometries of hyperbolic space are {\em relatively hyperbolic} relative to the parabolic subgroups.  For Veech groups, the space $D^{el}$ is quasi-isometric to the (hyperbolic) coned off Cayley graph illustrating (part of) the relative hyperbolicity of $G$.  We thus propose a kind of ``qualified'' notion of geometric finiteness with this in mind (the reason for the qualification is explained below).

\begin{definition} \label{D:par geom finite} A finitely generated group $G< \Mod(S)$ is {\em parabolically geometrically finite} if $G$ is relatively hyperbolic, relative to a (possibly trivial) collection of subgroups $\mathcal H = \{H_1,\ldots,H_k\}$ and
\begin{enumerate}
\item $H_i$ contains a finite index, abelian subgroup consisting entirely of multitwists, for each $1 \leq i \leq k$; and
\item The coned off Cayley graph $G$--equivariantly and quasi-isometrically embeds into $\C(S)$.
\end{enumerate}
\end{definition}

When $\mathcal H = \{ \{id\} \}$, we note that the condition is equivalent to $G$ being convex cocompact.  By Theorem~\ref{T:Tang}, lattice Veech groups are parabolically geometrically finite.  In fact, Tang's result is more general and implies that any finitely generated Veech group is geometrically finite by this definition.  These examples are all virtually free, but other examples include the combination subgroups of Leininger--Reid \cite{Leininger-Reid}, which are isomorphic to fundamental groups of closed surfaces of higher genus, and free products of higher rank abelian groups constructed by Loa \cite{Loa}.

In view of Theorem \ref{T:main simple}, one might formulate the following.

\begin{conjecture}
 Let $G< \Mod(S)$ be parabolically geometrically finite. Then the $\pi_1S$--extension group $\Gamma$ of $G$ is a hierarchically hyperbolic space.
\end{conjecture}

The reason for the ``qualification'' of the definition is that there are subgroups of $\Mod(S)$ that are candidates for being geometrically finite in some sense, but are not even relatively hyperbolic;  see the examples below. It is possible that there are different types of geometric finiteness for subgroups of mapping class groups, with Definition~\ref{D:par geom finite} being among the most restrictive. Other notions of geometric finiteness might include an HHS structure on the subgroup which is compatible with the ambient one on $\Mod(S)$ (e.g.~hierarchical quasiconvexity \cite{BHS:HHS2}).

Here are some candidate examples that may be considered geometrically finite:

\begin{itemize}
\item The whole group $\Mod(S)$. \item Multi-curve stabilizers.
 \item The right-angled Artin subgroups of mapping class groups constructed in \cite{CLM:raag_in_MCG, K:raag_in_MCG, runnels2020effective}.
 \item Free and amalgamated products of other examples.
\end{itemize}

\begin{question} For each example above, does the extension group admit an HHS structure?
\end{question}

We note that for the first example, the answer is `yes', since the extension group is the mapping class group of the puncture of $S$.

\bigskip

\noindent
{\bf Acknowledgments.}  The authors would like to thank MSRI and its Fall 2016 program on {\em Geometric Group Theory}, where this work began. We also gratefully acknowledge NSF grants DMS 1107452, 1107263, 1107367 (the GEAR Network) for supporting travel related to this project.
Dowdall was partially supported by NSF grants DMS-1711089 and DMS-2005368. Durham was partially supported by NSF grant DMS-1906487.  Leininger was partially supported by NSF grants DMS-1510034 and DMS-1811518.
Sisto  was partially supported by the Swiss National  Science Foundation (grant \#182186).

\section{Definitions and background} \label{S:Definitions}

We briefly recall some essential background material on coarse geometry, mapping class groups, Teichm\"uller spaces, quadratic differentials and Veech groups. As we expect that most readers will already be familiar with this material, we largely defer to other sources for details.

\subsection{Basic metric geometry}
A \emph{path} in a metric space $(X,d)$ is a continuous function $\gamma\colon [a,b]\to X$ from some (possibly degenerate) compact interval in the real line. Such a path \emph{connects} or \emph{joins} its endpoints $\gamma(a),\gamma(b)$ and has \emph{length}
\[\length(\gamma) = \sup \left\{\sum_{i=1}^{n} d(\gamma(t_{i-1}),\gamma(t_{i})) \mid a=t_0\leq \dots\leq t_n=b \right\}.\]
A path is \emph{rectifiable} its its length is finite, and is a \emph{geodesic} if its length equals the distance between its endpoints. The metric space is said to be geodesic if each pair of points in it are connected by a geodesic. It is customary to denote by $[x,y]$ any choice of a geodesic connecting $x$ to $y$. We will often conflate geodesics with their images, and same for paths.

A subset $Z\subset X$ is rectifiably path connected if any pair of points in $Z$ are connected by a rectifiable path $[a,b]\to Z$. In this case $Z$ inherits an \emph{induced path metric} where the distance between points $z_1,z_2\in Z$ is defined as the infimum of the lengths of all paths in $Z$ connecting $z_1$ and $z_2$.
We call $d$ a \emph{path metric} if it agrees with the induced path metric on $X$ itself; that is, if for all $x,y$ the distance $d(x,y)$ coincides with the infimum of the lengths of paths connecting $x$ to $y$.

Given two metrics $d$ and $d'$ on a set $X$, we say that $d$ is {\em coarsely bounded} by $d'$ if there exists a monotone function $N \colon [0,\infty) \to [0,\infty)$ so that $d(x,y) \leq N(d'(x,y))$, for all $x,y \in X$.  If $d$ is coarsely bounded by $d'$ and $d'$ is coarsely bounded by $d$, we say that $d$ and $d'$ are {\em coarsely equivalent}.

To any (set-theoretic) quotient $\pi\colon X\to Q$ of the metric space $(X,d)$, there is an associated \emph{quotient pseudo-metric} $d_Q$ on $Q$ defined as
\[d_Q(p,q)=\inf\sum_{i=1}^n d(x_i,y_i),\]
where the infimum is over all chains $x_1,y_1,\dots, x_n,y_n\in X$ satisfying $\pi(x_1)=p$, $\pi(x_n)=q$, and $\pi(y_i)=\pi(x_{i+1})$ for all $i=1,\dots,n-1$; see \cite[I.5.19]{BH:NPC}. Call such a chain \emph{tight} if $y_i \neq x_{i+1}$ for each $1\le i < n$. Observe that if a chain is not tight because $y_i = x_{i+1}$, then $ d(x_i, y_{i+1}) \le  d(x_i, y_i) +  d(x_{i+1},y_{i+1})$ and we may delete $y_{i},x_{i+1}$ to get a new chain with potentially smaller sum. Thus $d_Q(p,q)$ is in fact an infimum over tight chains.

\subsection{Graph approximation}
The $C \ge 0$ neighborhood of a subset $A$ of a metric space $(X,d)$ will be denoted
\[N_C(A) = \{x\in X \mid d(x,a) \le C\text{ for some }a\in A\}.\]
We say that $A$ is \emph{$C$--dense} if $X = N_C(A)$. A \emph{quasi-isometric embedding} $f\colon X\to Y$ is a map of metric spaces for which there exist constants $K\geq 1,C\geq 0$ such that
\[\frac{1}{K}d_Y(f(x),f(y)) -C \leq d_X(x,y)\leq Kd_Y(f(x),f(y))+C\]
for all $x,y\in X$.
The map is furthermore a \emph{quasi-isometry} if it is coarsely surjective, meaning that $f(X)$ is $R$--dense in $Y$ for some $R\ge 0$.
It is well known that every quasi-isometry $f\colon X\to Y$ admits a \emph{coarse inverse} quasi-isometry $g\colon Y\to X$ for which $g\circ f$ and $f\circ g$ are bounded distance from the identities on $X$ and $Y$. A quasi-isometric embedding with domain an interval in $\mathbb{R}$ or $\mathbb{Z}$ is called a \emph{quasi-geodesic}.

The following well-known fact will be useful in a few places.

\begin{prop}\label{P:graph approximation}
Suppose $\Omega$ is a path metric space and $\Upsilon \subset \Omega$ an $R$--dense subset, where $R > 0$. Fix $R'> 3R$ and consider a graph $\mathcal G$ with vertex set $\Upsilon$ such that:
  \begin{itemize}
   \item all pairs of elements of $\Upsilon$ within distance $3R$ are joined by an edge in $\mathcal G$,
   \item if an edge in $\mathcal G$ joins points $w,w'\in \Upsilon$, then $d_\Omega(w,w') \le R'$.
  \end{itemize}
 Then the inclusion of $\Upsilon$ into $\Omega$ extends to a quasi-isometry $\mathcal G \to \Omega$.
\end{prop}

\begin{proof}
We regard $\Upsilon$ as a subset of both $\Omega$ and $\mathcal G$, where we denote the respective metrics by $d_\Omega$, $d_{\mathcal G}$.
Let $f \colon \mathcal G \to \Omega$ be any map that includes the vertex set $\Upsilon$ into $\Omega$ and sends each edge to a constant speed path connecting the endpoints of length at most $R'$.  The map $f$ is clearly $R'$--Lipschitz and the image is $R$--dense.

To prove the remaining inequality, consider now arbitrary vertices $w,w'\in \Upsilon$ of $\mathcal G$.
Let $\alpha$ be a path connecting $w$ to $w'$ in $\Omega$ of length $Rk\leq d_\Omega(w,w')+R$ for some integer $k$. Subdividing $\alpha$ into subpaths of length $R$ and considering points of $\Omega$ within distance $R$ of the endpoints of the subpaths, we can find a sequence of points $w=w_0,\dots,w_k=w'$ of $\Upsilon$ so that $d_\Omega(w_i,w_{i+1})\leq 3R$. In particular,
$$d_{\mathcal G}(w,w')\leq k\leq d_\Omega(w,w')/R+1,$$
and the proof is complete.
\end{proof}

\subsection{Hyperbolicity}
A metric space $(X,d)$ is called Gromov \emph{hyperbolic} if there is a constant $\delta' \ge 0$ such that for all $x,y,z,w\in X$ one has
\[(x \vert z)_w \ge \min\big\{(x\vert y)_w, (y\vert z)_w\big\} - \delta',\]
where $(a \vert b)_c$ denotes $(d(a,c) + d(b,c) - d(a,b))/2$. In the case that $X$ is geodesic, this is equivalent to the existence of a constant $\delta\geq 0$ so that for all $x,y,z\in X$ and all choices of geodesics $[x,y],[y,z],[z,x]$ we have
\[[x,y]\subseteq N_\delta\big([y,z]\cup [z,x]\big).\]
When we want to specify the constant $\delta$, we say that the space $X$ is $\delta$--hyperbolic. We call $\delta$ a hyperbolicity constant for $X$.
It is a standard result that the property of being hyperbolic is preserved by quasi-isometries of path metric spaces; see e.g., \cite{vaisala-hyperbolic}.
It is also well-known that the hyperbolic plane $\mathbb H^2$ is, say, $2$--hyperbolic.

We will need the following criterion for hyperbolicity, which is an easy modification of one due to Masur--Schleimer \cite[Theorem 3.11]{MS:disks}, see also \cite[Proposition~3.1]{Bowditch:Hyp}.
For the statement,
given a metric space $\Omega$ and designated subsets $L(x,y)\subseteq \Omega$ for each pair $x,y\in \Omega$, we say that the subsets $L(x,y)$ \emph{form $\delta$--slim triangles} if for all $x,y,z\in\Omega$ we have $L(x,y)\subseteq N_\delta(L(x,z)\cup L(z,y))$. So, by definition, a geodesic metric space is hyperbolic if the set of geodesics form slim triangles.

\begin{prop}[Guessing geodesics]
\label{T:bowditch} Suppose $\Omega$ is a path metric space, $\Upsilon \subset \Omega$ an $R$--dense subset for some $R > 0$, and $\delta \ge 0$ a constant such that for all pairs $x,y \in \Upsilon$ there are rectifiably path-connected sets $L(x,y) \subset \Omega$ containing $x,y$ satisfying the properties:
\begin{enumerate}
\item the $L(x,y)$ form $\delta$--slim triangles, and
\item if $x,y \in \Upsilon$ have $d(x,y)\le 3R$, then the diameter of $L(x,y)$ is at most $\delta$.
\end{enumerate}
Then $\Omega$ is hyperbolic.
\end{prop}
\begin{proof}
For the special case of a graph $\mathcal G$ with $\Upsilon$ the $1$--dense set of vertices and the subsets $L(x,y)$ connected subgraphs, this statement is precisely the criterion of Bowditch \cite[Proposition~3.1]{Bowditch:Hyp}.
To translate this criterion for graphs into the condition we have here, we construct the graph $\mathcal G$ from Proposition~\ref{P:graph approximation} and prove that $\mathcal G$ is hyperbolic; this suffices since $\mathcal G$ is quasi-isometric to $\Omega$.

Let $g \colon \Omega \to \Upsilon$ be a map fixing $\Upsilon$ and assigning to $x\in \Omega$ some point of $\Upsilon$ within distance $R$ of $x$. Note that we will think of elements of $\Upsilon$ as contained both in $\mathcal G$ and in $\Omega$.
For $x,y \in \Upsilon$, let $\mathscr{L}(x,y)$ be the span of $g(L(x,y))$; that is, the largest subgraph whose vertex set is precisely $g(L(x,y))$.  Since $L(x,y)$ is rectifiably path connected, given any two points $w,w' \in L(x,y)$, there is a rectifiable path connecting them.  Partition this path into subpaths of length at most $R$, at points $w = w_0,w_1,\ldots,w_n = w'$.  Thus $d_\Omega(w_i,w_{i+1}) \leq R$, and so $d_\Omega(g(w_i),g(w_{i+1})) \leq 3R$, ensuring that $g(w_i)$ and $g(w_{i+1})$ are adjacent in $\mathcal G$.  Consequently, $w = w_0$ is connected by a path of length $n$ to $w' = w_n$ in $\mathcal G$ with all vertices in $g(L(x,y))$, and hence $\mathscr L(x,y)$ is connected.

Notice that $g$ is a quasi-isometry (since it is a coarse inverse of the quasi-isometry induced by inclusion from Proposition \ref{P:graph approximation}). Hence, since the $L(w,w')$ form $\delta$--slim triangles, we have that $\mathscr L(w,w')$ also form slim triangles (for a possibly larger constant).  Similarly, the bound on the diameter of the sets $L(w,w')$  implies a bound on the diameter of the sets $\mathscr L(w,w')$.  By the hyperbolicity criterion for graphs, it follows that $\mathcal G$, and hence $\Omega$, is hyperbolic.
\end{proof}

\subsection{Quasi-trees}

The following criterion for a graph to be a quasi-tree is well-known, and an easy consequence of Manning's bottleneck criterion \cite{Man:bottleneck}. We include a proof for completeness.

\begin{prop} \label{prop:bottleneck}
 Let $X$ be a graph, and suppose that there exists a constant $B$ with the following property. Suppose that for each pair of vertices $w,w'$ there exists an edge path $\gamma(w,w')$ from $w$ to $w'$ so that for any vertex $v$ on $\gamma(w,w')$, any path from $w$ to $w'$ intersects the ball of radius $B$ around $v$. Then $X$ is quasi-isometric to a tree, with quasi-isometry constants depending on $B$ only.
\end{prop}

\begin{proof}
 We check that \cite[Theorem 4.6]{Man:bottleneck} applies; that is, we check the following property.  For any two vertices $w,w' \in X$, there is a midpoint $m(w,w')$ between $w$ and $w'$ so that any path from $w$ to $w'$ passes within distance $B'=B'(B)$ of $m(w,w')$.  (The uniformity in the quasi-isometry comes from the proof of Manning's theorem, see \cite[page 1170]{Man:bottleneck}.)

 Consider any geodesic $\alpha$ from $w$ to $w'$, and let $m=m(w,w')$ be its midpoint. We will show that $m$ lies within distance $2B+1$ of a vertex of $\gamma=\gamma(w,w')$, so that we can take $B'=3B+1$.

Indeed, suppose by contradiction that this is not the case. Let $w=w_0,\dots,w_n=w'$ be the vertices of $\gamma$ (in the order in which they appear along $\gamma$), and let $d_i=d(w,w_i)$, so that $|d_{i+1}-d_i|\leq 1$. Each $w_i$ lies within distance $B$ of some point $p_i$ on $\alpha$ which must satisfy $d(p_i,m)\geq B+1$. In particular, we have that every $d_i$ satisfies either $d_i \leq d(w,w')/2 -1$ or $d_i\geq d(w,w')/2 +1$. Since $d_0=0$ and $d_n=d(w,w')$, we cannot have $|d_{i+1}-d_i|\leq 1$ for all $0\leq i\leq n-1$, a contradiction.
\end{proof}

\subsection{Surfaces and mapping class groups}
\label{S:Surface background}
\label{S:Mod S}
We next describe  a variety of geometric, topological, and analytic objects associated to a closed surface $S$ of genus at least two.
Firstly, we write $\Mod(S) = \pi_0(\Homeo^+(S))$ for its {\em mapping class group}, the group of components of its orientation preserving homeomorphisms.  Let $\dot S$ denote the surface $S$ with a marked point and $\Mod(\dot S)$ its mapping class group (in this case, all homeomorphisms must preserve the marked point).  Birman proved that by forgetting the marked point, these mapping class groups fit into the following {\em Birman exact sequence},
\[ 1 \to \pi_1S \to \Mod(\dot S) \to \Mod(S) \to 1.\]
Given a subgroup $G< \Mod(S)$, we let $\Gamma_G < \Mod(\dot S)$ denote the preimage in $\Mod(\dot S)$, which thus also fits into a short exact sequence
\begin{equation} \label{E:extension group} 1 \to \pi_1S \to \Gamma_G \to G \to 1
\end{equation}
mapping into the one above by inclusion. See e.g. \cite{Birman-book,Birman-MCGs,farb:MCG}.

\subsection{Teichm\"uller spaces} \label{S:Teich spaces}  Two complex structures $X$ and $Y$ on $S$ are equivalent if there is a biholomorphic map $(S,X) \to (S,Y)$ isotopic to the identity on $S$.  The {\em Teichm\"uller space} of $S$, denoted $\Teich(S)$, is the space of equivalence classes of complex structures on $S$.  Given a complex structure $X$ on $S$, we also write $X \in \Teich(S)$ for its equivalence class.  We equip $\Teich(S)$ with the Teichm\"uller metric, for which the distance from $X$ to $Y$ measures the maximal quasiconformal dilatation between the two complex structures at every point, minimized over all representatives of the equivalence classes.  See e.g. \cite{Gardiner,Imayoshi-Taniguchi, Bers60}.

Any complex structure $X$ on $S$ also determines a point of $\Teich(\dot S)$, where equivalence is defined via isotopies that fix the marked point.
Forgetting the marked point defines a fibration of Teichm\"uller spaces called the {\em Bers fibration},
\begin{equation} \label{E:bers fiber}
\widetilde S \to \Teich(\dot S) \to \Teich(S).
\end{equation}
Changing the complex structure on $\dot S$ by an isotopy that does {\em not} fix the base points gives different points in the fiber over $X \in \Teich(S)$.  The isotopy to the identity lifts to an isotopy to the identity of the universal covering $\widetilde S$, and tracking the location of a fixed lift of the marked point gives the identification of the fiber with $\widetilde S$.  The fibration is equivariant with respect to the homomorphisms in the Birman exact sequence.
See \cite{Bers-fiber_spaces,Farb-Mosher}.

\subsection{Quadratic differentials} \label{S:quad diffs} Given a complex structure $X$ on $S$, by a {\em quadratic differential for $X$} we mean a nonzero holomorphic section of the square of the canonical line bundle over $(S,X)$.  In a local coordinate $z$ for $X$, this is represented as $q(z) dz^2$, where $q$ is a holomorphic function.  Integrating the square root of a quadratic differential $q$ for $X$ (away from the zeros) gives a local coordinate $\zeta$ for $X$ called a {\em preferred coordinate for $q$} in which $q$ is represented as $d \zeta^2$.  The transition functions for overlapping preferred coordinates are locally given by $z \mapsto \pm z + c$ for some $c \in \mathbb C$.  The Euclidean metric, being invariant by such transformations, pulls back to a metric on the $S$ minus the zeros of $q$.  The metric completion with respect to this metric is a nonpositively curved Euclidean cone metric, called a {\em flat metric}, obtained by filling the zeros back in, so that there is a cone point of cone angle $k\pi$ at each zero of $q$ of order $k-2$.  We will write $q$ for both the quadratic differential as well as the associated flat metric.  See e.g. \cite{Gardiner,strebel:QD}.

\begin{remark} \label{R:sloppy differentials} Using the notation $q$ for both the quadratic differential as well as the flat metric is imprecise because the latter only determines the former up to multiplication by a unit modulus complex number.  When passing back and forth between the two, a choice of specific quadratic differential will either be irrelevant or the choice will be clear from the context.
\end{remark}

\subsection{Flat geodesics, foliations, and directions} \label{S:flat geodesics etc}

Consider a complex structure $X$ and flat metric $q$ (from a quadratic differential of the same name).  The geodesics for $q$ are Euclidean straight lines away from the cone points.  When a geodesic passes through a cone point, it subtends two angles, one on ``each side" of the cone point, each of which is at least $\pi$.  In fact, this characterizes $q$--geodesics: a path which is a Euclidean straight line away from the cone points, and makes angle at least $\pi$ on both sides of any cone point it meets is necessarily a geodesic.  Geodesic segments between pairs of cone points of $q$ that contain no cone points in their interior are called {\em saddle connections}.

Because the transition functions for the preferred coordinates defining the flat metric have the form $z \mapsto \pm z + c$, any line in the tangent space of a nonsingular point can be parallel translated around the surface, in the complement of the cone points, to produce a parallel line field.
Such line fields are in a one-to-one correspondence with the projective tangent space at any non-cone point, and we denote this space of directions as $\mathbb P^1(q)$.

Integrating a parallel line field gives a foliation by geodesics, which extends to a singular foliation over the entire surface. Thus for every $\alpha \in \mathbb P^1(q$), we have a singular foliation $\mathcal F(\alpha)$ in direction $\alpha$.  An important special case occurs when $\mathcal F(\alpha)$ has all nonsingular leaves being closed geodesics.  In this case, $S$ is a union of Euclidean cylinders, the interior of each foliated by parallel geodesic core curves in direction $\alpha$, and with boundary curves a geodesic concatenation of saddle connections (also in direction $\alpha$).  In this case, we say that $\mathcal F(\alpha)$ defines a {\em cylinder decomposition}.

\begin{example}
\label{ex:silver-L}  Consider the genus $2$ surface $S$ obtained from the polygon shown below with the sides identified by translations according to the numbering indicated. This surface is built from three squares and has an obvious $3$--fold branched cover map to the square torus.
\begin{center}
\begin{tikzpicture}
\draw[fill, opacity = .2] (0,0) -- (4,0) -- (4,2) -- (2,2) -- (2,4) -- (0,4) -- (0,0);
\draw (0,0) -- (4,0) -- (4,2) -- (2,2) -- (2,4) -- (0,4) -- (0,0);
\draw[line width = .1pt] (0,.2) -- (4,.2);
\draw[line width = .1pt] (0,.4) -- (4,.4);
\draw[line width = .1pt] (0,.6) -- (4,.6);
\draw[line width = .1pt] (0,.8) -- (4,.8);
\draw[line width = 1pt,dotted] (0,1) -- (4,1);
\draw[line width = .1pt] (0,1.2) -- (4,1.2);
\draw[line width = .1pt] (0,1.4) -- (4,1.4);
\draw[line width = .1pt] (0,1.6) -- (4,1.6);
\draw[line width = .1pt] (0,1.8) -- (4,1.8);
\draw (0,2) -- (2,2);
\draw[line width = .1pt] (0,2.2) -- (2,2.2);
\draw[line width = .1pt] (0,2.4) -- (2,2.4);
\draw[line width = .1pt] (0,2.6) -- (2,2.6);
\draw[line width = .1pt] (0,2.8) -- (2,2.8);
\draw[line width = 1pt,dashed] (0,3) -- (2,3);
\draw[line width = .1pt] (0,3.2) -- (2,3.2);
\draw[line width = .1pt] (0,3.4) -- (2,3.4);
\draw[line width = .1pt] (0,3.6) -- (2,3.6);
\draw[line width = .1pt] (0,3.8) -- (2,3.8);
\draw[fill] (0,0) circle (.05);
\draw[fill] (4,0) circle (.05);
\draw[fill] (4,2) circle (.05);
\draw[fill] (2,2) circle (.05);
\draw[fill] (2,4) circle (.05);
\draw[fill] (0,4) circle (.05);
\draw[fill] (0,2) circle (.05);
\draw[fill] (2,0) circle (.05);
\node[left] at (0,1) {$1$};
\node[left] at (0,3) {$2$};
\node[right] at (4,1) {$1$};
\node[right] at (2,3) {$2$};
\node[above] at (1,4) {$3$};
\node[below] at (1,0) {$3$};
\node[below] at (3,0) {$4$};
\node[above] at (3,2) {$4$};
\end{tikzpicture}
\end{center}
The Euclidean metric on the polygon descends to a flat metric on $S$ determined by a complex structure $X$ and quadratic differential $q$.  The vertices shown projects to a single cone point of cone angle $6\pi$.
The horizontal foliation illustrated defines a cylinder decomposition with core curves isotopic to the dotted and dashed curves and with boundaries made of the three horizontal saddle connections. In fact, any direction of rational slope defines a cylinder decomposition, as can be seen from the projection to the square torus.
\end{example}

The complex structure and quadratic differential can be pulled back to the universal covering $\widetilde S$, and will be denoted by the same names.  The metric on $\widetilde S$ is CAT(0) (since all cone angles are greater than $2\pi$), and in particular, any two points are connected by a unique geodesic segment.  The characterization of geodesics in $\widetilde S$ still holds, and saddle connections are defined similarly (or equivalently, as lifts of saddle connections).  The preimage of a cylinder on $S$ is an {\em infinite strip} in $\widetilde S$: this is an isometric embedding of a strip $[0,w] \times \mathbb R$ of some width $w > 0$.  The universal covering determines a canonical identification of the directions for $q$ on $\widetilde S$ with those on $S$, and we refer to either one as $\mathbb P^1(q)$.

\subsection{Teichm\"uller disks} \label{S:Teich disks defined} Suppose $q$ is any quadratic differential for $X$ with preferred coordinates $\{\zeta_j\}$ covering the complement of the zeros of $q$.  For any matrix $A \in \SL_2(\mathbb R)$, viewed as a real linear transformation of $\mathbb C$, we obtain a new atlas $\{A \circ \zeta_j\}$ defining a new complex structure and quadratic differential,
\[ A \cdot (X,q) = (A \cdot X,A \cdot q),\]
with the same set of zeros of the same orders.  (A caveat: the notation $A \cdot X$ only makes sense in the presence of a quadratic differential $q$ for $X$).  In this setting, the identity map $(S,X,q) \to (S,A \cdot X,A \cdot q)$ is {\em affine} in the respective preferred coordinates for $q$ and $A \cdot q$, and Teichm\"uller's theorem states that the distance between $X$ and $A \cdot X$ in $\Teich(S)$ is precisely the logarithm $\log(\|A\|)$ of the operator norm
\[ \|A\| = \max \{ \|A(v)\| \mid v \in \mathbb C, \|v\| = 1 \}.\]
We call $q$ and $A \cdot q$ the {\em initial} and {\em terminal flat metrics}, respectively.  We note that one usually refers to the specific quadratic differentials as initial and terminal, but this is unimportant for us; compare Remark~\ref{R:sloppy differentials}.
As the identity $(S,X,q) \to (S,A \cdot X,A \cdot q)$ is affine, it sends every $q$--geodesic to an $(A \cdot q)$--geodesic.

If $A \in \SO(2)$, then $X$ and $A \cdot X$ are equivalent (since a rotation is holomorphic), and so the map $A \mapsto A \cdot X$ defines a homeomorphism from the hyperbolic plane $\mathbb H^2=\SL_2(\mathbb R)/\SO(2)$ to the orbit $D_q\subset \Teich(S)$. By Teichm\"uller's theorem, the push-forward of the Poincar\'e metric is the restriction of the Teichm\"uller metric to $D_q$; in particular, $D_q$ is totally geodesic in $\Teich(S)$. We call $D_q$ the \emph{Teichm\"uller disk} of the quadratic differential $q$.  For any other point $(X',q') = A \cdot (X,q)$, we have $D_{q} = D_{q'}$ and we have a canonical identification $\mathbb P^1(q) \cong \mathbb P^1(q')$ from the affine identity map $id_S \colon (S,X,q) \to (S,X',q')$. See \cite{gardiner:QT} for details.

Any geodesic through $X$ in $D_q$ is given by $t \mapsto A_t \cdot (X,q)$, where $\{A_t\}_{t \in \mathbb R} < \SL_2(\mathbb R)$ is a symmetric, $1$--parameter, hyperbolic subgroup.  The (orthogonal) eigenlines for the common eigenvalues of the nontrivial elements give two direction in $\mathbb P^1(q)$, and we use this to identify the circle at infinity of $D_q$ with $\mathbb P^1(q)$.  Specifically, we identify the direction $\alpha$ of the contracting eigenline of $A_t$, $t > 0$ with the endpoint of the positive ray.  In particular, along this ray, the flat-lengths of any saddle connection or closed geodesic in direction $\alpha$ contracts exponentially.  Furthermore, the horocycle through $X$ based at the point at infinity associated to $\alpha$ is given by $t \mapsto B_t \cdot (X,q)$, where $\{B_t\}_{t \in \mathbb R} < \SL_2(\mathbb R)$ is a $1$--parameter parabolic subgroup with all nontrivial elements having common eigenline $\alpha$.  Consequently, a saddle connection or closed geodesic in direction $\alpha$ has constant length along this horocycle.  Combining these two properties, we have the following useful fact.
\begin{proposition} \label{P:horoball sublevel set}
Given a quadratic differential $q$ for some complex structure $X$ on $S$, and any saddle connection or simple closed geodesic $\sdl$ with respect to $q$ in direction $\alpha \in \mathbb P^1(q)$, the horoballs (respectively, horocycles) based at $\alpha$ in $D_q$ are sublevel sets (respectively, level sets) of the length of $\sdl$. \qed
\end{proposition}

\subsection{Veech groups}  \label{S:Veech groups} Given a quadratic differential $q$ for $X$ and associated Teichm\"uller disk $D_q \subset \Teich(S)$, the stabilizer of $D_q$, denoted $G_q < \Mod(S)$ is called the {\em Veech group} of $q$ (or $D_q$).   This is equivalently the subgroup of $\Mod(S)$ consisting of all mapping classes represented by homeomorphisms that are affine in preferred coordinates for $q$.  The derivative of any such affine homeomorphism in preferred coordinates is well defined, up to sign, and defines a homomorphism $G_q \to \PSL_2(\mathbb R)$.  The action of $G_q$ on $D_q$ gives a homomorphism from $G_q \to \Isom^+(D_q) \cong \PSL_2(\mathbb R)$, and up to conjugation, this is precisely the derivative homomorphism.  Since the action of $\Mod(S)$ on $\Teich(S)$ is properly discontinuous, so is the action of $G_q$ on $D_q$, and hence the image in $\PSL_2(\mathbb R)$ is discrete.  We say that $(S,X,q)$ is a \emph{lattice surface} if $G_q$ is a lattice (i.e.~the quotient $D_q/G_q$ has finite area).

An element of $G_q$ is called {\em elliptic}, {\em parabolic}, or {\em hyperbolic}, respectively, if its image under the derivative map is of that type.  Elliptic elements fix a point of $D_q$, while parabolic and hyperbolic elements $g \in G_q$ fix one or two points, respectively, in $\mathbb P^1(q)$.  If $g$ is parabolic with fixed point $\alpha\in \mathbb P^1(q)$, then the foliation $\mathcal F(\alpha)$ defines a $g$--invariant cylinder decomposition, and a power of $g$ is a composition $\tau_\alpha$ of Dehn twists in the core curves of the cylinders. This power of $g$ is the identity on the saddle connections in direction $\alpha$, which form a union of spines for the complementary components of the core curves of the cylinders.  The set of {\em parabolic directions}, denoted $\mathcal P(q) \subset \mathbb P^1(q)$, is the fixed points of parabolic elements in $G_q$.

Given $\alpha \in \mathcal P(q)$, the boundary of every cylinder in direction $\alpha$ consists of saddle connections.  The key result for us in the next theorem is that all saddle connections arise in this way in the case of interest to us. See \cite{Masur-Tabachnikov,Thurston-geomDynSurfs} for details.

\begin{theorem} [Veech dichotomy]
\label{T:Veech_Dichotomy}
Suppose $(S,X,q)$ is a lattice surface.  Then every saddle connection is contained in the boundary of a cylinder in direction $\alpha \in \mathcal P(q)$.  Moreover, for every $\alpha \not \in \mathcal P(q)$, every half leaf of $\mathcal F(\alpha)$ is dense in $S$.
\end{theorem}

\begin{example}
Consider the Veech group $G_q$ for the surface $(S,X,q)$ of Example~\ref{ex:silver-L}.
There are two parabolic elements $g,h \in G_q$ with derivatives
\[ dg = \left( \begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array} \right) \quad \mbox{ and } \quad dh = \left( \begin{array}{cc} 1 & 0 \\ 2 & 1 \end{array} \right).\]
The element $g$ preserves the horizontal foliation and is isotopic to a composition of Dehn twists about the core curves of the corresponding cylinder decomposition.
The element $h$ similarly preserves the vertical foliation and is given by a composition of Dehn twists.
The derivatives of the elements $g,h \in G_q$ generate a lattice in $\PSL_2(\mathbb R)$.  It follows that $\langle g,h \rangle < G_q$ is a finite index subgroup, and $(S,X,q)$ is a lattice surface. Since the rational slopes are precisely the directions of cylinder decompositions, we see from the Veech dichotomy that $\CP(q)=\mathbb P^1(\mathbb Q)$ in this case.
\end{example}

\section{Setup and Notation}
\label{S:setup}

We now fix a flat metric (i.e.~quadratic differential) $q$ for a complex structure $X_0$ in the Teichm\"uller space $\Teich(S)$ of a closed surface $S$ of genus at least $2$, and assume $(S,X_0,q)$ is a lattice surface.  Let $D = D_q \subset \Teich(S)$ be the Teichm\"uller disk of $q$ and let $\rho$ be the Poincar\'e metric on $D$.
For any $X \in D$, let $q_X$ be the terminal flat metric from the Teichm\"uller mapping $(S,X_0,q) \to (S,X,q_X)$.
Let $G = G_q \le \Mod(S)$ be the lattice Veech group of $q$ and  $\CP = \CP(q) \subset \mathbb{P}^1(q)$ be the set of parabolic directions on $q$, which are precisely the directions of the saddle connections by Theorem~\ref{T:Veech_Dichotomy}.

\subsection{Horoballs}
For each direction $\alpha\in \CP$, we fix a closed horoball $B_\alpha\subset D$ that is invariant by the corresponding parabolic subgroup of $G$. We choose these so that the family $\{B_\alpha\}_{\alpha\in \CP}$ is $G$--invariant and \emph{$1$--separated}, meaning that the pairwise distance between any two horoballs is at least $1$. For each $\alpha\in \CP$, we let $c_\alpha\colon D\to B_\alpha$ be the $\rho$--closest-point projection map. This map is $1$--Lipschitz, as can be seen by observing that the derivative of $c_\alpha$ is norm non-increasing at each point $X\in D$.

Let $p \colon D \to \hat D$ denote the quotient space obtained by collapsing each $B_\alpha$ to a point, and give $\hat D$ the quotient pseudo-metric $\hat \rho$.  It is straightforward to see that this is indeed a metric and in fact, according to \cite[Lemma I.5.20]{BH:NPC}, a path metric.  The $1$--separated assumption implies that $\hat D$ is quasi-isometric to the electric space obtained by coning each $B_\alpha$ to a point, and that $p(B_\alpha)$ and $p(B_{\alpha'})$ are at least $\hat\rho$--distance $1$ apart if $\alpha \neq \alpha'$.

We also consider the associated truncated Teichm\"uller disk
\[ \bar D = D \setminus \bigcup_{\alpha \in \CP} B_\alpha^\circ,\]
where $B_\alpha^\circ$ is the interior of $B_\alpha$. The induced path metric $\bar \rho$ on $\bar D$ is CAT(0) by \cite[Theorem II.11.27]{BH:NPC}, and the group $G$ acts cocompactly on $\bar D$.
The projections $c_\alpha$ above now restrict to maps $c_\alpha \colon \bar D \to \partial B_\alpha \subset \bar D$ that remain $1$--Lipschitz.

\subsection{Bundles and the total space}\label{sec:bundles-total-space} Let $\pi \colon E \to D$ be the pull-back bundle of the Bers fibration \eqref{E:bers fiber} via the inclusion $D \subset \Teich(S)$.
We identify $E \subset \Teich(\dot S)$.
Let $\Gamma = \Gamma_G  < \Mod(\dot S)$ be the extension of $G$ as in \eqref{E:extension group}.
The group $\Gamma$ acts on $E$ and the quotient $E/\Gamma$ is a noncompact $S$--bundle over $D/G$.
To rectify this noncompactness, we also consider the pull-back bundle over $\bar D$
\[ \bar E = \pi^{-1}(\bar D) = E \setminus \bigcup_{\alpha \in \CP} \CB_\alpha^\circ,\]
where $\CB_\alpha^\circ = \pi^{-1}(B_\alpha^\circ)$ is the interior of the horoball preimage $\CB_\alpha = \pi^{-1}(B_\alpha)$ for $\alpha\in \CP$.
The space $\bar E$ is a $4$--manifold with boundary
\[ \partial \bar E = \pi^{-1}\Big(\bigcup_{\alpha \in \CP} \partial B_\alpha \Big) = \bigcup_{\alpha \in \CP} \partial \CB_\alpha,\]
and the group $\Gamma$ acts on $\bar E$ with quotient $\bar E/\Gamma$ a compact $S$--bundle over $\bar D/G$:
\[ S \to \bar E/\Gamma \to \bar D/G.\]

\subsection{Moving between fibers}
\label{sec:moving_between_fibers}
For any $X \in D$ let $E_X = \pi^{-1}(X) \subset E$ be the fiber over $X$, with $E_0 = E_{X_0}$ denoting the fiber over $X_0$.  The fiber $E_X$ is canonically identified with the universal cover of $S$ equipped with the (pulled back) complex structure $X$ and flat metric $q_X$.

Given $X,Y \in D$, let $f_{X,Y} \colon E_Y \to E_X$ be the lift of the Teichm\"uller mapping to the universal covering, which is affine with respect to the flat metrics $q_X$ and $q_Y$ on the domain and range, respectively.  We note that $f_{X,Y} f_{Y,Z} = f_{X,Z}$ and that $f_{X,Y}$ is $e^{\rho(X,Y)}$--biLipschitz (in preferred Euclidean coordinates, $f_{X,Y}$ is given by $(u,v) \mapsto (e^{\rho(X,Y)}u,e^{-\rho(X,Y)}v$). Varying over all $Y \in D$, these determine a map
\[f_X \colon E \to E_X,\quad\text{where}\quad f_X\vert_{E_Y} = f_{X,Y}\quad\text{for any $Y \in D$}.\]
In the case of $X_0$, we simply write $f = f_{X_0} \colon E \to E_0$.  For any $X \in D$ and $x \in E_X$, the fiber $D_x = f_X^{-1}(x)$ is the unique Teichm\"uller disk in $\Teich(\dot S)$ through $x$ that covers $D$ via the projection $\pi$.  We also consider these objects in $\bar E$ over $\bar D$, as well, writing $\bar D_x = D_x \cap \bar E$, which is a truncated Teichm\"uller disk mapping bijectively to $\bar D$.

For any $\alpha \in \CP$, let $f_\alpha \colon \bar E \to \partial \CB_\alpha \subset \bar E$ be defined by
\[ f_\alpha(x) = f_{c_\alpha(\pi(x))}(x),\]
for any $x \in \bar E$.  In words, $f_\alpha$ restricted to any fiber $E_X$ is the canonical map $f_{Y,X}$ to the fiber $E_Y$, where $Y = c_\alpha(X)$ is the $\rho$--closest point on $\partial B_\alpha$ to $X$.

\subsection{Cone points}
For $X \in D$, let $\Sigma_X \subset E_X$ denote the set of cone points of the flat structure $q_X$ on $E_X$. We write $\Sigma_0 = \Sigma_{X_0}$, as a special case, and define
\[ \Sigma = \bigcup_{X \in D} \Sigma_X \quad \mbox{ and } \quad \bar \Sigma = \Sigma \cap \bar E.\]
Note that $\Sigma$ is the union of all Teichm\"uller disks through all cone points and $\bar \Sigma$ the union of truncated Teichm\"uller disks through cone points.

\subsection{Bundle metrics}
\label{sec:bundle_metrics}
We give $E$ a metric $d$ defined as the flat metric in each fiber $E_X$, and orthogonal Poincar\'e metric $\rho$ in each Teichm\"uller disk $D_x$.  Alternatively, in $E - \Sigma$, the metric is Riemannian given by the orthogonal direct sum of the flat metric in the fiber and Poincar\'e metric on Teichm\"uller disks, and then $E$ can be identified as the metric completion.  This metric is invariant by $\Gamma$.

The subspace $\bar E \subset E$ is given the induced path metric which we denote $\bar d$.
Since $\Gamma$ acts isometrically and cocompactly on $(\bar E, \bar d)$, this metric space will serve as our quasi-isometric model for the extension group $\Gamma$.

We note that $\bar \Sigma$ is $r$--dense for some $r > 0$, which follows from the compactness of the quotient metric space $\bar E/\Gamma$.
Using the Arzel\`a--Ascoli Theorem and the compactness of $\bar E / \Gamma$, it is easy to see that any sequence of paths joining $x$ and $y$ in $\bar E$ whose lengths converge to $\bar d(x,y)$ has a subsequential limit that is, necessarily, a geodesic. Thus $(\bar E, \bar d)$ is a geodesic metric space.

From the $G$--invariant quotient $p \colon D \to \hat D$ we construct a $\Gamma$--invariant quotient $P \colon E \to \hat E$ as follows.
For each $\alpha$, fix some point $X_\alpha \in \partial B_\alpha$.
Let $T_\alpha$ be the (simplicial) $\mathbb R$--tree dual to the foliation of $E_{X_\alpha}$ in the direction $\alpha$ (i.e.~a weighted Bass--Serre tree) with metric defined by the transverse measure on the foliation of $E_{X_\alpha}$ in direction $\alpha$. Then  let
\[ t_\alpha \colon E \to T_\alpha \]
be the composition of the projection $E_{X_\alpha} \to T_\alpha$ with the map $f_{X_\alpha} \colon E \to E_{X_\alpha}$.

Note that the map $t_{\alpha}$ is independent of our choice of point $X_\alpha \in \partial B_\alpha$.  We also comment that there is a uniform lower and upper bound of the length of every edge of every tree $T_{\alpha}$ since there are only finitely-many $\Gamma$--orbits of edges.

\begin{definition}[Maximal hyperbolic space of the hierarchy]
Let $P \colon E \to \hat E$ denote the quotient obtained by collapsing each set $\CB_\alpha$ onto $T_\alpha$ via the map $t_\alpha|_{\CB_\alpha}$.
Equip $\hat E$ with the quotient pseudo-metric $\hat d$ obtained from $\bar d$ under the (surjective) restriction
\[\bar P = P\vert_{\bar E} \colon \bar E \to \hat E.\]
\end{definition}

Lemma~\ref{L:quotient metric on E hat} below shows that $\hat d$ is in fact a path metric. One of the main results of this paper, Theorem~\ref{T:hyperbolicity of hat E}, is that the metric space $\hat E$ is hyperbolic.

The map $\bar  \pi = \pi\vert_{\bar E}\colon \bar E\to \bar D$ descends to a quotient map $\hat \pi \colon \hat E \to \hat D$ that is $1$--Lipschitz  by construction (as it is the descent of a $1$--Lipschitz map).
For every $x \in E$, the image $\hat D_x$ of $D_x$ in $\hat E$ is a obtained by collapsing $\CB_\alpha \cap D_x$ to a point, for each $\alpha \in \CP$, and hence $\hat \pi|_{\hat D_x} \colon \hat D_x \to \hat D$ is a bijection.  In particular, each $\hat D_x$ with its path metric is isometric to $\hat D$, and isometrically embedded in $\hat E$.
Objects in $E$, $\bar E$, and $\hat E$ are called {\em vertical} if they are contained in a fiber of $\pi$, $\bar \pi$, or $\hat \pi$, respectively, and {\em horizontal} if they are contained in $D_x$, $\bar D_x$, or $\hat D_{x}$, for some $x \in E,\bar E$.
The following commutative diagram
summarizes the situation:
\[
\begin{tikzcd}[row sep=6]
                 & E \ar[dr,  "P"] \ar[dd, near start, "\pi"] \\
\bar E \arrow[ur, hook] \ar[dd,  "\bar \pi"] \ar[rr,  near start,  "\bar P" below, crossing over] &&  \hat E \ar[dd,  "\hat \pi"]\\
                 & D \ar[dr,  "p"] \\
\bar D \arrow[ur, hook] \ar[rr,  "\bar p"] &&  \hat D\\
\end{tikzcd}
\]

\begin{lemma} \label{L:quotient metric on E hat}
The quotient pseudo-metric $\hat d$ on $\hat E$ is a path metric.  The map $\bar P \colon \bar E \to \hat E$ is $1$--Lipschitz and is a local isometry at every point $x \in \bar E - \partial \bar E$.  Furthermore, for every $\alpha\in \CP$,
\begin{itemize}
\item The induced path metric on $P(\partial \CB_\alpha) = T_\alpha$ is the $\mathbb{R}$--tree metric determined by the transverse measure on the foliation of $E_{X_\alpha}$ in direction $\alpha$.
\item The subspace topology on $T_\alpha \subset \hat E$ agrees with the $\mathbb{R}$--tree topology on $T_\alpha$.
\end{itemize}
\end{lemma}
\begin{proof}
The map $\bar P$ is $1$--Lipschitz by definition of $\hat d$.
According to \cite[Lemma I.5.20]{BH:NPC}, since $\bar d$ is a path metric, $\hat d$ is necessarily a path metric provided it is a metric.
Recall that for $\hat x,\hat y\in \hat E$, the quotient pseudo-metric $\hat d(\hat x, \hat y)$ is the infimum of $\sum_i \bar d(x_i,y_i)$
over all tight chains $x_1,y_1,\dots,x_n,y_n$ from $\hat x$ to $\hat y$, meaning that $\bar P(x_1) = \hat x$, $\bar P(y_n) = \hat y$, and $\bar P(y_i) = \bar P(x_{i+1})$ for all $1\le i < n$ with $y_i \neq x_{i+1}$.

First suppose $x\in \bar E \setminus \partial \bar E$ and set $3\epsilon = \bar d(x, \partial \bar E) > 0$. Let $U\subset \bar E$ be the ball of radius $\epsilon$ about $x$ and let $z\in U$ be arbitrary. For any $y\in \bar E$, consider any tight chain $z_1,y_1\dots,z_n,y_n$ from $\hat z = \bar P(z)$ to $\hat y =\bar P(y)$. If $y_1\ne y$, then tightness implies $y_1\in \partial \bar E$ (since $\bar P$ is injective on $\bar E\setminus \partial \bar E$) and hence $\sum_i \bar d(z_i,y_i) \ge \bar d(z_1,y_1) \ge 2\epsilon$. Otherwise we have the trivial chain $z=z_1,y_1=y$ with sum $\bar d(z,y)$. This proves
\[\bar d(z,y) \ge \hat d(\hat z,\hat y) \ge \min\{2\epsilon, \bar d(z,y)\}\quad\text{for all $z\in U$ and $y\in \bar E$}.\]
In particular, $\hat d(\hat z,\hat y) = \bar d (z,y)$ for all $z,y\in U$, proving that $\bar P$ is a local isometry on $\bar E\setminus \partial \bar E$. We also see that $\hat d(\hat z,\hat y) = 0$ forces $z=y$ and hence $\hat z = \hat y$, establishing the positive definiteness of $\hat d$ in the case that one point lies in $\bar P(\bar E\setminus \partial \bar E)$.

For $\alpha \in \CP$, let $\bar \ell_\alpha$ be the path metric on $\partial \CB_\alpha \subset \bar E$, and $\ell_\alpha$ the path pseudo metric on $T_\alpha \subset \hat E$. Also let $d_\alpha$ be the $\mathbb R$--tree metric on $T_\alpha$ and note that  $\ell_\alpha \le d_\alpha$ by construction. For any $X\in \partial B_\alpha$, the restriction of $\bar P\vert_{\partial \CB_\alpha} = t_\alpha \vert_{\partial \CB_\alpha}$ to $E_X$ is just the projection $E_X\to T_\alpha$ onto the $\mathbb R$--tree dual to the foliation of $E_X$ in direction $\alpha$.
Since the restriction of $\bar d$ to $\partial \CB_\alpha$ is the orthogonal sum of flat length in fibers and horocycle length in Teichm\"uller disks, we see that every path $\gamma$ in $\partial \CB_\alpha$ satisfies
\[\bar \ell_\alpha\text{--}\mathrm{length}(\gamma)\ge d_\alpha\text{--}\mathrm{length}(\bar P(\gamma)).\]

Next suppose $x\in \partial \bar E$, and say $x\in \partial \CB_\alpha$. Since $\hat \pi \colon \hat E \to \hat D$ is $1$--Lipschitz, we have $\hat d(\bar P(x),\bar P(y)) \ge \hat \rho (\hat \pi (\bar P(x)),\hat \pi(\bar P(y))) > 0$ for any $y\notin \partial \CB_\alpha$. For the purposes of proving $\hat d$ is a metric, it therefore suffices to fix some $0 < \epsilon < \tfrac{1}{2}$ and consider a point $y\in \partial \CB_\alpha$ such that $\hat d(\bar P(x),\bar P(y)) < \epsilon$.
Take any tight chain $x_1,\dots,y_n$ from $\hat x = \bar P(x)$ to $\hat y = \bar P(y)$ with $\sum_{i}\bar d(x_i,y_i) < \hat d(\hat x,\hat y)+\epsilon < 2\epsilon$. By tightness, $x_i,y_i\in \partial \bar E$ for each $1\le i \le n$. If $x_i$ and $y_i$ lie in distinct components of $\partial \bar E$, then $\bar d(x_i,y_i) \ge \bar \rho(\bar \pi(x),\bar\pi(y)) \ge 1$ by the fact that our horoballs $\{B_\alpha\}_{\alpha \in \CP}$ are $1$--separated. As this contradicts the assumption on $\sum_i \bar d(x_i,y_i)$, we must have $x_i,y_i\in \partial \CB_\alpha$ for all $1\le i \le n$.
Now let $\gamma_i$ be a geodesic joining $x_i$ to $y_i$ in $\bar E$ of length $\bar d(x_i,y_i) < 2\epsilon$. Hence $\gamma_i$ lies in the $\epsilon$--neighborhood of $\partial \CB_\alpha$. The map $f_\alpha \colon \bar E\to \partial \CB_\alpha$ is $e^{\epsilon}$--Lipschitz on this neighborhood, as can be seen by noting that this holds in the fiber direction (since $f_{X,Y}$ is $e^{\rho(X,Y)}$--Lipschitz) and $c_\alpha$ is $1$--Lipschitz. Therefore $f_\alpha(\gamma_i)$ is a path in $\partial \CB_\alpha$ of length at most $e^{\epsilon} \bar d(x_i,y_i)$. Since $\bar P(f_\alpha(y_i)) = \bar P(f_\alpha(x_{i+1}))$, the images $\bar P(f_\alpha(\gamma_i))$ concatenate to give a path $\gamma$ in $T_\alpha$ from $\hat x$ to $\hat y$ satisfying
\begin{align*}
\hat d(\hat x,\hat y) &\le \ell_\alpha(\hat x, \hat y) \le d_\alpha(\hat x, \hat y)\le d_\alpha\text{--}\mathrm{length}(\gamma)\\
& = \sum_{i=1}^n d_\alpha\text{--}\mathrm{length}(\bar P(f_\alpha(\gamma_i))) \le e^{\epsilon} \sum_{i=1}^n \bar d(x_i,y_i).
\end{align*}
Since this holds for all tight chains, we conclude that $e^{-\epsilon}d_\alpha(\hat x, \hat y)$ is a lower bound on $\hat d(\hat x, \hat y)$ whenever $\hat x,\hat y\in T_\alpha$ satisfy $\hat d(\hat x, \hat y) < \epsilon$. This proves $\hat d$ is a metric, since we now see that $\hat d(\hat x,\hat y) = 0$ implies $d_\alpha(\hat x,\hat y) = 0$ and hence $\hat x = \hat y$.

Since any path in $T_\alpha$ may be subdivided into pieces whose successive endpoints satisfy $\hat d(\hat x_i,\hat x_{i+1})< \epsilon$, this also proves that $\ell_\alpha \ge e^{-\epsilon}d_\alpha$ for all small $\epsilon > 0$. Therefore $\ell_\alpha = d_\alpha$ as claimed by the lemma. Finally, this argument establishes Lipschitz inequalities $e^{-\epsilon} d_\alpha \le \hat d \le d_\alpha$ for nearby points in $T_\alpha$ and, specifically, proves that for each $\hat x\in T_\alpha$ and all small $\epsilon > 0$ we have
\[\big\{\hat y \in T_\alpha \mid d_\alpha (\hat x,\hat y)<\epsilon \big\}
 \subset \big\{\hat y\in T_\alpha \mid \hat d(\hat x, \hat y) < \epsilon \big\}
 \subset \big\{\hat y \in T_\alpha \mid d_\alpha (\hat x, \hat y) < e^\epsilon \epsilon \big\}.\]
Thus the $\mathbb{R}$--tree and subspace topologies on $T_\alpha$ agree, and the lemma holds.
\end{proof}

\begin{lemma} \label{L:P is also 1-Lip} The map $P \colon E \to \hat E$ is $1$--Lipschitz.
\end{lemma}
\begin{proof} Suppose $x,y \in E$ are any two points.  Given any $\epsilon > 0$, there is a path $\gamma$ from $x$ to $y$ with length $(1+\epsilon)d(x,y)$ that decomposes as a concatenation $\gamma = \gamma_1 \cdots \gamma_k$, with each $\gamma_i$ contained in $\bar E$ or in $\CB_\alpha$, for some $\alpha \in \mathcal P$.   Since $\epsilon$ is arbitrary, it suffices to prove that the length of $P(\gamma)$ is no greater than the length of $\gamma$.  If $\gamma_i$ is a path in $\bar E$, then the $d$--length is equal to the $\bar d$--length (since $\bar d$ is the path metric induced by $d$), and since $\bar P$ is $1$--Lipschitz, the length of $P(\gamma_i)$ is no greater than the length of $\gamma_i$.  On  the other hand, if $\gamma_i$ is contained in $\CB_\alpha$, then it maps by $P$ to the tree $T_\alpha$.  The restriction of $P$ to each fiber $E_X$, for $X \in B_\alpha$ is $1$--Lipschitz, and the horizontal directions collapse completely, and so it is easy to see that the length of $P(\gamma_i)$ is no more than that of $\gamma_i$, thus completing the proof.  \end{proof}

\subsection{Tree Vertices}
\label{S:tree-vertices}
Let $\vtx \subset \hat E$ be the set of all vertices of all trees $T_\alpha$ over all $\alpha \in \CP$. We view $\vtx$ both as a subset of $\hat E$ and abstractly as an indexing set that will be used in sections \S\ref{S:projections}--\ref{S:combinatorial HHS} to develop an HHS structure on $\bar E$.  We write $\alpha \colon \vtx \to \CP$ to denote the map that associates to $v$ the direction $\alpha(v) \in \CP$ so that $v \in T_{\alpha(v)}$.
Given $v \in \vtx$, we will write $B_v = B_{\alpha(v)}$, $\partial B_v = \partial B_{\alpha(v)}$, etc and write $c_v = c_{\alpha(v)}$ for the $\rho$--closest point projection $D \to B_v$.

For $v,w \in \vtx$, we write $v \parallel w$ if $\alpha(v) = \alpha(w)$. Then define  $d_{\tree}(v,w) \in \mathbb Z_{\geq 0} \cup \{\infty\}$ to be the combinatorial distance in the simplicial tree $T_{\alpha(v)}= T_{\alpha(w)}$ when $v \parallel w$  (as opposed to the distance from the $\mathbb R$--tree metric) and to
equal $\infty$ when $v\not\parallel w$.

\subsection{Spines}
\label{S:spines}
For any $\alpha \in \CP$ and $X \in D$, we consider the union of the set of all saddle connections in direction $\alpha$ in $E_X$.  The components of this space are precisely the preimages of vertices of $v \in T_\alpha$ under the map $P \circ f_{X_\alpha,X} \colon E_X \to T_\alpha$.  For a vertex $v \in T_\alpha^{(0)}\subset \vtx$ and $X \in D$, we write $\theta^v_X = (P \circ f_{X_\alpha,X})^{-1}(v) \subset E_X$ for this component, which we call the {\em $v$--spine in $E_X$}.  The closure of each component of
\[ E_X -  \bigcup_{v \in T_\alpha^{(0)}} \theta^v_X \]
is an infinite strip covering a Euclidean cylinder in direction $\alpha$.  For each $v$, let $\bTheta^v_X$ be the union of $\theta^v_X$ together with all these infinite strips that meet $\theta^v_X$, which we call the {\em thickened $v$--spine in $E_X$}.  Note that $f_{X,Y}$ maps $\theta^v_Y$ and $\bTheta^v_Y$, respectively, to $\theta^v_X$ and $\bTheta^v_X$, respectively.

The union of the $v$--spines and thickened $v$--spines over $\partial B_v$ are denoted
\[ \theta^v = \bigcup_{X \in \partial B_v} \theta^v_X \quad \quad \bTheta^v = \bigcup_{X \in \partial B_v} \bTheta^v_X. \]
These are bundles over $\partial B_v$ which we call the {\em $v$--spine bundle} and the {\em thickened $v$--spine bundle}.

\begin{lemma}
\label{L:strip-and-saddle-bound}
There exists a constant $M > 0$ such that
\begin{enumerate}
\item For every $X\in \bar D$, every saddle connection in $E_X$ has length at least $\frac{1}{M}$.
\item For each $v\in \vtx$ and $X\in \partial B_v$, every saddle connection in $\theta^v_X$ has length at most $M$ and every strip in $\bTheta^v_X$ has width at most $M$. In particular, for points $X\in \partial B_\alpha$, the saddle connections and strips of $E_X$ in direction $\alpha\in \CP$ have, respectively, uniformly bounded lengths and widths.
\item For every $X \in \bar D$, there is a triangulation of $E_X$ by saddle connections so that the triangles have diameters at most $M$ and interior angles at least $\frac1M$.

\end{enumerate}
\end{lemma}
\begin{proof}
For each $X \in D$, the fiber $E_X$ is canonically identified with the universal cover of the closed surface $S$ equipped with the flat metric for the quadratic differential $q_X$.  The existence of a (finite) triangulation of $(S,X,q_X)$ by saddle connections follows from \cite{Masur-Smillie}, and we lift this to a triangulation of $E_X$.

Since $(S,X,q_X)$ has finitely many cone points, there is a lower bound on the flat distance between any two cone points and hence a lower bound on the length of any saddle connection on $E_X$.  For each $\alpha\in \CP$, the flat surface $(S,X,q_X)$ decomposes into finitely many cylinders in direction $\alpha$ whose boundary curves are geodesic concatenations of saddle connections in direction $\alpha$. In particular, $(S,X,q_X)$ has only finitely many saddle connections in the $\alpha$ direction. Since the strips and saddle connections in the $\alpha$ direction on $E_X$ are precisely the preimages of these finitely many cylinders and saddle connections on $(S,q_X)$, there is a maximal width/length of any strip/saddle connection on $E_X$ in the $\alpha$ direction.

Items (1) and (3) follow from compactness of $\bar D / G$. Item (2) follows from the facts that widths/lengths of strips/saddle connections in direction $\alpha$ over a $\partial B_\alpha$ are constant (Proposition~\ref{P:horoball sublevel set}) and that there are only finitely many $G$--orbits in $\CP$.
\end{proof}

\subsection{A useful diagram}

Figure \ref{fig:Veech bundle} collects many of the main pieces of the setup for the paper.  The truncated Teichm\"uller disk $\bar {D}$ is the base of the $\mathbb{H}^2$--bundle $\bar{E}$.  The rest of the pieces include:
\begin{figure}
\includegraphics[width=\linewidth]{Veechbundle.jpg}
\caption{The general setup for the paper.}
\label{fig:Veech bundle}
\end{figure}
\begin{itemize}
\item[(a)] Three $1$--separated horoballs.

\item[(b)]  A horoball in the direction $\beta$ with the chosen basepoint $X_{\beta}$.  Twisting via the multitwist $\tau_{\beta}\in G$ moves $X_{\beta}$ along the horocycle.

\item[(c)] Two fibers in the universal cover, $\partial \mathcal B_{\beta}$, of the graph manifold which is the bundle over the horocycle in direction $\beta$.  The fibers $E_{X_{\beta}}$ and $E_{\tau_{\beta}(X_{\beta})}$ are above $X_{\beta}$ and $\tau_{\beta}(X_{\beta})$, respectively.  The multitwist $\tau_{\beta}$ induces shearing in the fibers.

\item[(d)] A horoball in the direction $\alpha$, with the chosen basepoint $X_{\alpha}$ and its horocycle $\partial B_{\alpha}$.

\item[(e)] The fiber $E_0$ over the basepoint $X_0$.  A flat geodesic $\gamma$ between cone points in the fiber $E_0$ is a concatenation of saddle connections.  The fiber $E_0$ maps to the fiber $E_{X_{\alpha}}$ by the lift $f_{X_{\alpha}}$ of the Teichm\"uller mapping between $X_0$ and $X_{\alpha}$.

\item[(f)] The fiber $E_{X_{\alpha}}$ over the horocyclic point $X_{\alpha}$.  The spine $\theta^v_{X_{\alpha}}$ in direction $\alpha$ is in red, with the thickened spine neighborhood $\bTheta^v_{X_{\alpha}}$ indicated in lavender.  Other spines stick off from and form the boundary of $\bTheta_{X_{\alpha}}$.

\item[(g)] The Bass--Serre tree $T_{\alpha}$ for the direction $\alpha$ with the map $t_{\alpha}\colon E_{X_{\alpha}} \to T_{\alpha}$.  The various spines are collapsed to the vertices and two spines are connected by an edge in $T_{\alpha}$ if their neighborhoods meet.
\end{itemize}

\subsection{Combinatorial paths and distances}
\label{S:combinatorial distances}

In studying $\bar E$ and $\hat E$, it will be helpful to utilize certain well-behaved paths that, in particular, allow us to understand when pairs of points are bounded distance apart.

\begin{lemma} \label{L:combinatorial paths in bar E}
There exists $R > 0$ so that any two points $x,y \in \bar \Sigma$ are connected in $\bar E$ by a path of length at most $R \bar d(x,y)$ that is a concatenation of at most $R\bar d(x,y)+1$ pieces, each of which is either a saddle connection of length at most $R$ in a vertical vertical fiber, or a horizontal geodesic segment in $\bar E$.
\end{lemma}
\begin{proof}
Take a geodesic $\gamma'$ in $\bar E$ joining $x,y\in \bar \Sigma$. Let $L\le \bar d(x,y)$ be the length of the projected path $\bar \pi(\gamma')$ in $\bar D$, and choose an integer $n$ so that $n-1\le L < n$. Divide $\gamma'$ into $n$ subpaths $\gamma' = \gamma'_1\dotsb\gamma'_n$ such that $\bar \pi(\gamma'_i)$ has length $L/n\le 1$.  Let $x'_{i-1},x'_i$ be the endpoints of $\gamma'_i$ and set $X_i = \pi(x'_i)\in \bar D$. For each $0 < i < n$, the diameter bound  for a triangulation as in Lemma~\ref{L:strip-and-saddle-bound} allows us to connect $x'_i\in E_{X_i}$ to some cone point $x_i\in \Sigma_{X_i}$ by a vertical path of length at most $M$. Append these to the endpoints of $\gamma'_i$ to obtain a path $\gamma_i$ from $x_{i-1}$ to $x_i$ with $\bar\pi(\gamma_i)= \bar\pi(\gamma'_i)$. These concatenate to give a path $\gamma = \gamma_1\dotsb \gamma_n$ in $\bar E$ joining $x$ to $y$ of total length
\[\mathrm{length}(\gamma) \le \bar d(x,y) + 2M(n-1) \le \bar d (x,y) + 2ML \le (2M+1)\bar d(x,y).\]

For each $0 < i \le n$, set $y_i = f_{X_{i},X_{i-1}}(x_{i-1})$ and connect $x_{i-1}$ to $y_i$ by a horizontal geodesic $h_i$. Note that $h_i$ has length at most that of $\bar \pi(\gamma'_i)$.  Since $\bar \rho(X_{i-1},X_{i})\le 1$ by construction, $f_{X_{i},X_{i-1}}$ is $e$--Lipschitz. Therefore $\mu'_i = f_{X_i,X_{i-1}}(\gamma_i)$ a  path in $E_{X_i}$ whose length is at most $e$ times the length of $\gamma_i$. Pushing $\mu'_i$ into the $1$--skeleton of the triangulation of $E_{X_i}$ produces a new path $\mu_i$, whose length grows by another fixed factor, that connects $y_i$ to $x_i$ via sequence of vertical saddle connections of length at most $M$.

Putting it all together, we have a path $h_1\mu_1\dotsb h_n\mu_n$ from $x$ to $y$ consisting of horizontal geodesics $h_i$ and vertical saddle connections of length at most $M$. The horizontal pieces contribute total length
\[\sum_{i=1}^n \bar d\text{--}\mathrm{length}(h_i) \le \sum_{i=1}^n \bar \rho\text{--}\mathrm{length}(\bar\pi(\gamma'_i)) \le \bar d\text{--}\mathrm{length}(\gamma') = \bar d(x,y).\]
Similarly, the vertical pieces contribute total length at most a fixed multiple of $\bar d\text{--}\mathrm{length}(\gamma) \le (2M+1)\bar d(x,y)$. Hence the length of $h_1\mu_1\dotsb h_n\mu_n$ is bounded as desired. Since each saddle connection has length bounded below (Lemma~\ref{L:strip-and-saddle-bound}), the number of saddle connections in this concatenation is linearly bounded by $\bar d(x,y)$, and the number of horizontal pieces is $n \le L+1 \le \bar d(x,y) + 1$.
\end{proof}

\begin{lemma} \label{L: collapsed vertex set dense}
There exists $R_0 > 0$ so that $\vtx$ is $R_0$--dense in $\hat E$.
\end{lemma}
\begin{proof} This follows from the fact that $\vtx$ is $\Gamma$--invariant and that $\hat E/\Gamma$ is compact, being the continuous image of $\bar E/\Gamma$ under the descent of $\bar P \colon \bar E \to \hat E$.
\end{proof}

\begin{definition} \label{D:combinatorial path}
A \emph{horizontal jump} in $\hat E$ is the image under $\bar P$ of a geodesic in $\bar D_z$, for some $z\in \bar\Sigma$,
that connects two components of $\partial \bar D_z$ and whose interior is disjoint from $\partial \bar D_z$.
A \emph{combinatorial path} in $\hat E$ is a concatenation of horizontal jumps.
\end{definition}

Note that every horizontal jump in $\hat E$ connects points of $\vtx$, by construction, and has length at least $1$. Indeed, for distinct $\alpha,\alpha'\in \CP$, the points $\hat \pi(P(\CB_{\alpha}))$ and $\hat \pi(P(\CB_{\alpha'}))$ have distance at least $1$. Since $\hat \pi$ is $1$--Lipschitz, any path joining $\partial \CB_{\alpha}$ to $\partial \CB_{\alpha'}$ in $\bar E$ thus projects to a path of length at least $1$ in $\hat E$. In particular, the number of jumps in a combinatorial path is bounded by the path's total length.

\begin{lemma}  \label{L: combinatorial path}
There is constant $C>0$ such that
any pair of points $x,y\in \vtx$ may be connected by a combinatorial path of length at most $C\hat d(x,y)$ that, in particular, consists of at most $C\hat d(x,y)$ horizontal jumps.
\end{lemma}

\begin{proof}
For $\alpha \in \CP$, let $\bar \ell_\alpha$ be the path metric on $\partial \CB_\alpha \subset \bar E$. Also let  $\ell_\alpha$ be the path metric on $T_\alpha \subset \hat E$ which, recall from Lemma~\ref{L:quotient metric on E hat}, is the $\mathbb R$--tree metric dual to the foliation of $E_{X_\alpha}$.
Finally, let $\bar \Sigma_\alpha = \bar\Sigma\cap \partial \CB_\alpha$ and note that there exists some $K\ge 30$, independent of $\alpha\in \CP$, such that each $z\in \partial \CB_\alpha$ satisfies $\bar d(z,w)\le K/30$ for some $w\in \bar \Sigma_\alpha$; indeed by Lemma~\ref{L:strip-and-saddle-bound}(3) we may take $K=30M$.
As a first step towards the lemma, we show how to jump between points of $\bar P(\bar \Sigma_\alpha)$:

\begin{claim}
\label{claim:combinatorial_in_tree}
There exists $R'>0$ such that any pair of points $v_1,v_2\in \bar P(\bar \Sigma_\alpha)\subset \vtx$ may be connected by a combinatorial path of length at most $R' \ell_\alpha(v_1,v_2)$.
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:combinatorial_in_tree}]
By decomposing a $T_\alpha$--geodesic from $v_1$ to $v_2$ into its edges, it suffices to suppose $v_1$ and $v_2$ are adjacent vertices of $T_\alpha$. Pick any point $X\in \partial B_\alpha$ and consider the restriction $\bar P\vert_{E_X}\colon E_X \to T_\alpha$. The preimage of $v_1,v_2$ under this map are adjacent spines $\theta^{v_1}_X,\theta^{v_2}_X$ which are separated by a strip whose width is bounded by Lemma~\ref{L:strip-and-saddle-bound}. Therefore we may pick a bounded-length saddle connection $\sigma \subset E_X$ joining cone points $y_1\in \theta^{v_1}_X$ to $y_2 \in \theta^{v_2}_X$.

Let $\beta \in \CP$ be the direction of $\sigma$. Since $\sigma$ has bounded length, $X$ is bounded distance in $D$ from some point $Y$ in the horocycle $\partial B_\beta$. Let $z_i = f_{Y,X}(y_i)$ and let $h_i$ be the horizontal geodesic in $D_{y_i}$ from $y_i$ to $z_i$. By definition, the $\bar P$--image of each component of $h_i\cap \bar D_{y_i}$ is a horizontal jump in $\hat E$. Each of these jumps has length at most $\rho(X,Y)$ which itself is uniformly bounded. The saddle connection $f_{Y,X}(\sigma)$ in $E_Y$ is collapsed to a point by $\bar P$. Hence taking the jumps from $h_1$ and then coming back along the jumps from $h_2$ gives a bounded length combinatorial path in $\hat E$ from $v_1$ to $v_2$. Since $\ell_\alpha(v_1,v_2)$ is uniformly bounded below (by the minimal length of an edge in $T_\alpha$) the claim follows.
\end{proof}

We now prove the lemma in general. Fix $x,y\in \vtx$. If $x=y$ the claim is vacuous, so assume $r = \hat d(x,y)>0$. Let $x_1,y_1,\dots,x_n,y_n\in \bar E$ be any tight chain from $\bar P(x)$ to $\bar P(y)$, as in Lemma~\ref{L:quotient metric on E hat}, with $\sum_{i=1}^n \bar d(x_i,y_i) \le 2 r$. By tightness, and the fact $\bar P(x_1),\bar P(y_n)\in \vtx$, we have $x_i,y_i\in \partial \bar E$ for all $1\le i \le n$.
We note, moreover, that $y_i,x_{i+1}$ lie in a common boundary component $\partial \CB_{\alpha}$ for each $1\le i<n$.

We call a pair $x_i,y_i$ along this chain a {skip} (in contrast to a jump) if $x_i\in \partial \CB_\alpha$ and $y_i\in \partial \CB_{\beta}$ for distinct directions $\alpha,\beta \in \CP$.
Observe that in this case $\bar d(x_i,y_i) \ge 1$ since the horoballs $B_\alpha,B_\beta$ are $1$--separated.
Choose nearby points $w_i\in \bar \Sigma_\alpha$ and $z_i\in \bar\Sigma_\beta$ so that $\bar d(x_i,w_i),\bar d(z_i,w_i)\le K/30$. Therefore
\[\bar d(x_i,w_i) + \bar d(w_i,z_i) + \bar d(z_i,y_i) \le \frac{4K}{30} + \bar d(x_i,y_i) \le  \frac{K}{6}\bar d(x_i,y_i).\]
Now insert copies of $w_i,z_i$ to get a new chain $\dots,x_i,w_i,w_i,z_i,z_i,y_i,\dots$ (of $2(n+2)$ elements) where the skip $w_i,z_i$ is between points of $\bar \Sigma\cap \partial\bar E$. Making an insertion for each skip and relabeling, as necessary, we henceforth assume our chain $x_1,\dotsc,y_n$ has $\sum_{i=1}^n \bar d(x_i,y_i) \le Kr/3$, lies in $\partial \bar E$, and only skips between points of $\bar \Sigma$.

Consider again a skip $x_i,y_i$ of our improved chain, say with  $x_i\in \bar\Sigma_\alpha$. Let $Z_i\in \bar D$ be the closest point on $\partial B_\alpha$ to $\bar \pi(y_i)$, and let $z_i\in \partial \CB_\alpha\cap \bar D_{y_i}\subset \bar\Sigma_\alpha$ be its lift to the Teichm\"uller disk $\bar D_{y_i}$.
By construction of $\bar d$, the $\bar \rho$--geodesic from $\bar \pi(y_i)$ to $Z_i = \bar \pi(z_i)$ lifts to a path from $y_i$ to $z_i$ in $\bar E$ with the same length. Since any path $\gamma'$ from $y_i$ to $\partial \CB_\alpha$ has
\[\mathrm{length}(\gamma')\ge \mathrm{length}(\bar \pi(\gamma')) \ge \bar \rho (\bar \pi(z_i)), \bar \pi(y_i)) = \bar d (y_i,z_i),\]
we see that $\bar d(z_i, y_i) \le \bar d(x_i,y_i)$. Therefore, by the triangle inequality, we have
\[\bar d(x_i,z_i) + \bar d(z_i,y_i) \le \bar d(x_i,y_i) + 2\bar d(y_i,z_i) \le 3 \bar d(x_i,y_i).\]
This means we may insert $z_i$ to get a new chain $\dots,x_i,z_i,z_i,y_i,\dots$ whose distance sum is at most $Kr$. Making such an adjustment for each skip and relabeling if necessary, we may now assume that our chain $x_1,\dots,y_n$ lies in $\partial \bar E$, that $\sum_{i=1}^n \bar d(x_i,y_i) \le Kr$, and that each skip is horizontal and between points of $\bar \Sigma$, meaning that $x_i,y_i\in \bar \Sigma$ lie in the same Teichm\"uller disk $\bar D_{x_i} = \bar D_{y_i}$ whenever they lie in distinct components of $\partial \bar E$.

Now suppose that $\alpha \in \CP$ and $1 \le j \le k \le n$ are such that the points $x_j,y_j,\dots,x_k,y_k$ all lie in $\partial \CB_\alpha$. For each $j \le i \le k$, the pair $x_i,y_i$ may be joined by a path $\gamma_i$ in $\bar E$ of length at most $2\bar d(x_i,y_i) \le 2Kr$. Hence $\gamma_i$ is contained in the $Kr$--neighborhood of $\partial \CB_\alpha$ in $\bar E$. Since the map $f_\alpha \colon \bar E \to \partial \CB_\alpha$ is $e^{Kr}$--Lipschitz on this set  and restricts to the identity of $\partial \CB_\alpha$, we conclude $f_\alpha(\gamma_i)$ has length at most $2e^{Kr}\bar d(x_i,y_i)$.
Since $\bar P$ is $1$--Lipschitz, the images of the paths $f_\alpha(\gamma_i)$ under $\bar P$ therefore concatenate to yield a path in $T_\alpha$ from $\bar P(x_j)$ to $\bar P(x_k)$ of length at most
\[\sum_{i=j}^k \mathrm{length}(f_\alpha(\gamma_i)) \le 2e^{Kr} \sum_{i=j}^k \bar d(x_i,y_i).\]
Since $\bar P(x_k),\bar P(x_j)\in \bar P(\bar \Sigma_\alpha)$, we may now use Claim~\ref{claim:combinatorial_in_tree} to construct a combinatorial path from $\bar P(x_j)$ to $\bar P(x_k)$ of length at most $2R' e^{Kr}\sum_{i=k}^j \bar d(x_i,y_i)$.

On the other hand, for each skip $x_i,y_i$ in our chain the horizontal geodesic in $\bar D_{x_i} = \bar D_{y_i}$ from $x_i$ to $y_i$ projects to a combinatorial path from $\bar P(x_i)$ to $\bar P(y_i)$ of length at most $\bar d(x_i,y_i)$. Concatenating these with the  paths produced above for each maximal subchain $x_j,y_j,\dots,x_k,y_j$ in some common component $\partial\bar \CB_\alpha$, we finally produce a combinatorial path from $x = \bar P(x_1)$ to $y = \bar P(y_n)$ of length at most $2R'e^{Kr}\sum_{i=1}^n \bar d(x_i,y_i) \le 4R' e^{Kr}r$.

We now show lemma holds for $C = 36 R' e^{3KR_0}$, where $R_0$ is the density constant from Lemma~\ref{L: collapsed vertex set dense}. If $r=\hat d(x,y) \le 3R_0$, then the above produces a combinatorial path from $x$ to $y$ of length at most $\frac{C}{9} \hat d (x,y)$, as desired. If $r > 3 R_0$, then we may join $x$ to $y$ by a path $\gamma$ of length at most $2 r$. Subdivide $\gamma$ into $n = \lceil \length(\gamma)/R_0\rceil$ subsegments of equal length at most $R_0$. By Lemma~\ref{L: collapsed vertex set dense}, each subdivision point is within distance $R_0$ of $\vtx$. In this way, we obtain a sequence $x = x_0,\dots,x_n =y$ in $\vtx$ with $\hat d(x_i, x_{i+1})\le 3R_0$ for each $i$. Connecting each $x_i$ to $x_{i+1}$ by a combinatorial path of length at most $\frac{C}{3}R_0$, we obtain a combinatorial path from $x$ to $y$ of length at most $\frac{C}{3}R_0 n$. Since $R_0 n < \length(\gamma) + R_0 \le 3\hat d(x,y)$, we are done.
\end{proof}

\section{Hyperbolicity of $\hat E$} \label{S:slim triangles}

The goal of this section is to prove the following.

\begin{theorem} \label{T:hyperbolicity of hat E} The space $\hat E$ is hyperbolic.
\end{theorem}

This is achieved by applying the hyperbolicity criterion given by Proposition \ref{T:bowditch} (Guessing geodesics) to the collection of  (collapsed) \emph{preferred paths}---a special type of combinatorial path as in Definition~\ref{D:combinatorial path}---which we now describe.

\subsection{Preferred paths} \label{S:preferred paths}

For any two points $x,y \in \Sigma$, our next goal is to construct a particular path $\varsigma(x,y)$ from $x$ to $y$. Recall from \S\ref{sec:bundle_metrics} that for each $\alpha \in \CP$ we have fixed a point  $X_\alpha\in B_\alpha$.
To begin, we connect the points $f(x),f(y) \in E_0$ by a geodesic segment $[f(x),f(y)]$ in the flat metric $q$ of $E_0$.  This geodesic is a concatenation of saddle connections in $E_0$,
\[ [f(x),f(y)] = \sdl_1 \sdl_2 \cdots \sdl_k, \]
where the saddle connection $\sdl_i$ has direction $\alpha_i \in \CP$.
For each $1 \leq i \leq k$, let $z_{i-1},z_i$ be the initial and terminal endpoints, respectively, of $\sdl_i$.

Roughly speaking, $\varsigma(x,y)$ is the path that starts at $x$, follows a horizontal geodesic in $E$ to the fiber $E_{X_{\alpha_1}} \subset \partial \mathcal B_{\alpha_1}$ where saddle connections in direction $\alpha_1$ are short.  The path then traverses the saddle connection $\sdl_1$ in $E_{X_{\alpha_1}}$, then continues on along a horizontal geodesic to the fiber $E_{X_{\alpha_2}} \subset \partial \mathcal B_{\alpha_2}$ and traverses the next saddle connection $\sdl_2$.  The path continues in this way traversing horizontal geodesic segments followed by short saddle connections, as dictated by $[f(x),f(y)]$, until all of the saddle connections $\sigma_i$ have been traversed and the path can go to $y$ along a horizontal geodesic.

More formally, we let $\varsigma(x,y)$ be the concatenation of segments
\[ \varsigma(x,y) = h_0\gamma_1h_1 \gamma_2 h_2 \cdots \gamma_kh_k \]
defined as follows.
For each $i = 1,\ldots,k$, $\gamma_i$ is the saddle connection
\[ \gamma_i = f_{X_{\alpha_i}}(\sdl_i)  \subset E_{X_{\alpha_i}}.\]
The paths $h_i$ are horizontal geodesic segments making $\varsigma(x,y)$ into a path.  More precisely, let $\gamma_i^-,\gamma_i^+$ denote the initial and terminal endpoints of $\gamma_i$, respectively, and observe that for $i = 1,\ldots,k-1$, we have
\[ \gamma_i^+,\gamma_{i+1}^- \in D_{z_i}, \quad x,\gamma_0^- \in D_{z_0}, \mbox{ and } \, \, y,\gamma_k^+ \in D_{z_k}.\]
Then for $i = 1,\ldots,k-1$, $h_i$ is the geodesic from $\gamma_i^+$ to $\gamma_i^-$ in $D_{z_i}$, and $h_0$ and $h_k$ are the geodesic segments from $x$ to $\gamma_0^-$ and $\gamma_k^+$ to $y$ in $D_{z_0}$ and $D_{z_k}$, respectively.
We call the segments $\gamma_i$ the {\em saddle pieces} and the $h_i$ the {\em horizontal pieces}.   We note that when $\gamma_i$ and $\gamma_{i+1}$ are in the same direction, then $h_i$ is degenerate.
Also observe that the construction is symmetric: $\varsigma(x,y)$ and $\varsigma(y,x)$ are the same paths with opposite orientations.

We call $\varsigma(x,y)$ the {\em preferred path} from $x$ to $y$.
We can push preferred paths forward via the $1$--Lipschitz map $P\colon E\to \bar E$ (Lemma \ref{L:P is also 1-Lip}) and thus consider the image $\hat \varsigma(x,y) = P(\varsigma(x,y))$.
We will call $\hat \varsigma(x,y)$ a {\em collapsed preferred path}.  Note that collapsed preferred paths are combinatorial paths.
The key fact about these paths needed to prove that $\hat E$ is hyperbolic is the following.

\begin{theorem} \label{T:preferred slim triangles} There exists $\delta > 0$ so that collapsed preferred paths form $\delta$--slim triangles.  That is, for any $x,y,z \in \Sigma$, we have
\[ \hat \varsigma(x,y) \subset N_\delta(\hat \varsigma(x,z)\cup\hat \varsigma(y,z)).\]
\end{theorem}

We divide the proof of this theorem into a sequence of lemmas, which requires some further setup and notation, and occupies the bulk of this section.
Before we do that, however, we assume Theorem~\ref{T:preferred slim triangles} and use it to prove Theorem~\ref{T:hyperbolicity of hat E}.

\subsection{Hyperbolicity}
Given $u,v \in \vtx$, define
\[ L(u,v) =\bigcup \hat \varsigma(x,y),\]
where the union is taken over all $x \in \theta^u\cap \Sigma$ and $y \in \theta^v \cap \Sigma$.
Since $\varsigma(x,y)$ is a finite length path connecting $x$ and $y$ and $P$ is Lipschitz,
it follows that $L(u,v)$ is path connected, rectifiable set containing both $u$ and $v$.

\begin{lemma} \label{L:L(u,v) slim} Suppose $u,v \in \vtx$. For any $x \in \theta^u \cap \Sigma$ and $y \in \theta^v\cap \Sigma$ we have
\begin{equation}\label{E:just one varsigma} L(u,v) \subset N_{2\delta}(\hat \varsigma(x,y))
\end{equation}
where $\delta > 0$ is the constant from Theorem~\ref{T:preferred slim triangles}.
\end{lemma}
\begin{proof}
Let $x' \in \theta^u \cap \Sigma$ and $y' \in \theta^v \cap \Sigma$ be any points.
Since $[f(x),f(x')]$ is a concatenation of saddle connections all in direction $\alpha(u)$ and since $x,x' \in \mathcal B_{\alpha(u)}$, we have $\hat \varsigma(x,x') = \{u\}$.  Similarly, $\hat \varsigma(y,y')=\{v\}$.
Now by Theorem~\ref{T:preferred slim triangles}, we have
\[ \hat \varsigma(x',y) \subset N_\delta(\hat \varsigma(x,y) \cup \hat \varsigma(x,x')) = N_\delta(\hat \varsigma(x,y)),\]
and
\[ \hat \varsigma(x',y') \subset N_\delta(\hat \varsigma(x',y) \cup  \hat \varsigma(y,y')) = N_\delta(\hat \varsigma(x',y)) \subset N_{2 \delta}(\hat \varsigma(x,y)).\]
Since $x',y'$ were arbitrary, the result follows.
\end{proof}

Recall from Lemma~\ref{L: collapsed vertex set dense} that $\vtx$ is $R_0$ dense in $\hat E$.
\begin{lemma} \label{L:L(u,v) bdd diam} There exists a constant $C >0$ so that if $u,v \in \vtx$ with $\hat d(u,v) \leq 3R_0$, then $\diam(L(u,v)) \leq C$.
\end{lemma}
\begin{proof} Let $ x \in \theta^u \cap \Sigma$ and $y \in \theta^v \cap \Sigma$.  By Lemma~\ref{L: combinatorial path}, there is a bounded length combinatorial path connecting $P(x)=u$ to $P(y)=v$ which is a concatenation of $n$ horizontal jumps, where the bound on the length and the number $n$ depends only on $R_0$.  Each such horizontal jump is a collapsed preferred path.

By iterated application of Theorem~\ref{T:preferred slim triangles} one can show that $\hat \varsigma (x,y)$ is contained in the $n\delta$--neighborhood of this combinatorial path, and hence has uniformly bounded diameter.
\end{proof}

The proof of Theorem \ref{T:hyperbolicity of hat E} is now an easy consequence:

\begin{proof}[Proof of Theorem~\ref{T:hyperbolicity of hat E}]  Since $\vtx \subset \hat E$ is $R_0$ dense by Lemma~\ref{L: collapsed vertex set dense}, we need only verify the sets $L(u,v)$ satisfy the two conditions of Proposition~\ref{T:bowditch}.  Theorem~\ref{T:preferred slim triangles} and Lemma~\ref{L:L(u,v) slim} immediately imply the sets $L(u,v)$ form $3\delta$--slim triangles, verifying condition (1), and Lemma~\ref{L:L(u,v) bdd diam} precisely gives condition (2).
\end{proof}

We now record a corollary of Theorem \ref{T:hyperbolicity of hat E}:

\begin{corollary}\label{cor:Cayley_graph}
Let $\Upsilon_1,\ldots,\Upsilon_k < \Gamma$ be representatives of the conjugacy classes of vertex subgroups, and let $\mathcal S$ be any finite generating set for $\Gamma$. Then the Cayley graph $Cay(\Gamma,S\cup\bigcup \Upsilon_i)$ is $\Gamma$--equivariantly quasi-isometric to $\hat{E}$, and in particular it is hyperbolic.
\end{corollary}

\begin{proof}
 This follows from Theorem \ref{T:hyperbolicity of hat E} and the fact that the Cayley graph described in the statement is quasi-isometric to $\hat E$ by the version of the Schwartz-Milnor Lemma given by \cite[Theorem 5.1]{CharneyCrisp}.

 In fact, the action of $\Gamma$ on $\hat{E}$ is cocompact since the action of $\Gamma$ on $\bar E$ is, and the natural map from $\bar E$ to $\hat{E}$ is continuous. Point-stabilizers for the action on $\hat E$ are either trivial or conjugate into one of the subgroups described in the statement. Finally, discreteness of orbits follows considering separately points outside of the $T_\alpha$ (which have neighborhoods isometric to open sets of $\bar E$), vertices of the $T_\alpha$, and points along edges of the $T_\alpha$, in the latter two cases appealing to Lemma \ref{L:quotient metric on E hat}.
\end{proof}

We now move on to the analysis of triangles of collapsed preferred paths necessary to prove Theorem~\ref{T:preferred slim triangles}.

\subsection{(Non)degenerate triangles}
The proof of Theorem \ref{T:hyperbolicity of hat E} relies on an analysis of geodesic triangles in the base fiber $E_0$ with cone point vertices.  Given any $x,y,z \in \Sigma$, write
\[ \Delta(x,y,z) = \Delta(f(x),f(y),f(z)) = [f(x),f(y)] \cup [f(y),f(z)] \cup [f(z),f(x)] \]
for the associated geodesic {\em reference triangle} in $E_0$.
Such a triangle may have some {\em degenerate corners} where the initial geodesic segments emanating from a vertex share a number of saddle connections.  See Figure \ref{F:general triangle}.  As illustrated in the figure, there is a subtriangle $\Delta' \subset \Delta$, any two distinct edges of which intersect in exactly one point (an endpoint). The fact that geodesic triangles in $E_0$ actually fit this description is easily deduced from the fact that the flat metric on $E_0$ is uniquely geodesic, as it is CAT(0). If $\Delta = \Delta'$, then we say that $\Delta$ is {\em nondegenerate}, and otherwise it is {\em degenerate}.

Let $\Delta^\varsigma(x,y,z) \subset E$ be the triangle of preferred paths and $\Delta^{\hat \varsigma}(x,y,z) \subset \hat E$ its image in the collapsed space.  We say that these triangles are degenerate or nondegenerate according to whether $\Delta(x,y,z)$ is.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale = .5]
\draw[ultra thick] (0,0) --  (5,1) -- (10,-1) -- (7,1) -- (6,2) -- (5,6) -- (4.5,4) -- (3,1.5) -- (0,0);
\draw (5,6) -- (5,6.5) -- (4.5,7) -- (5,8);
\draw (10,-1) -- (10.5,-1.25) -- (11,-.5) -- (11.5,-1) -- (12,-1);
\draw (0,0) -- (-.5,0) -- (-1,.5) -- (-1.5,0) -- (-2,0) -- (-2.5,-.5) -- (-3,-.5);

\end{tikzpicture}
\caption{A general triangle $\Delta$ in $E_0$.  The nondegenerate subtriangle $\Delta'\subset \Delta$ is shown in bold.}
\label{F:general triangle}
\end{center}
\end{figure}

\begin{lemma} \label{L:excising degenerate parts}  If there exists $\delta > 0$ so that every nondegenerate triangle $\Delta^{\hat \varsigma}(x,y,z)$ is $\delta$--slim, then the same is true for degenerate triangles of collapsed preferred paths.
\end{lemma}
In particular, to prove Theorem~\ref{T:preferred slim triangles}, it suffices to prove that nondegenerate triangles of collapsed preferred paths are $\delta$--slim.
\begin{proof}  Assume all nondegenerate triangles of collapsed preferred paths are $\delta$--slim for some $\delta$, and let $x,y,z \in \Sigma$ be any points.  If $[f(x),f(y)] \cap [f(x),f(z)] = \sdl_1 \cdots \sdl_r$, where $\sdl_i$ is a saddle connection, then the preferred paths from $x$ to $y$ and from $x$ to $z$ agree on the first $r$ horizontal pieces and $r$ saddle pieces.  Likewise for the segments starting at $y$ and $z$.  Points along the common paths are within distance $0$ of another side.  After excising these common paths, we are left with a nondegenerate triangle of preferred paths, which is $\delta$--slim by assumption.  In particular, the distance from any point on one side of the original triangle either has distance $0$ or distance at most $\delta$ from some point on one of the other sides, thus the original (arbitrary) triangle is $\delta$--slim.
\end{proof}

\subsection{Decomposing triangles}
To prove Theorem~\ref{T:preferred slim triangles},  we can restrict our attention to points $x,y,z$ for which the reference triangle $\Delta(x,y,z)$ in the base fiber $E_0$ is nondegenerate, by Lemma~\ref{L:excising degenerate parts}.  Our analysis of the collapsed preferred paths of nondegenerate triangles is then carried out in steps by analyzing increasingly complicated reference triangles.
We now prove two lemmas which will be useful for decomposing more complicated triangles in the base fiber into simpler pieces.

\begin{lemma} \label{L:no cones inside} Given cone points $x,y,z \in \Sigma_0$, if $\Delta(x,y,z)$ is nondegenerate, then it bounds a topological $2$--simplex (also denoted $\Delta(x,y,z)$) which is convex and has no cone points in its interior.
\end{lemma}
\begin{proof} To see that $\Delta(x,y,z)$ bounds a convex topological 2-simplex, we notice that we can obtain the simplex by prolonging the sides to geodesic lines, and intersecting half-spaces bounded by these lines.

Let $N\ge 0$ denote the number of cone points in the interior of $\Delta = \Delta(x,y,z)$, each of which has cone angle at least $3\pi$. The double $\Delta'$ of $\Delta$ along its boundary is a topological sphere. If $a,b,c>0$ are the interior angles of the vertices of $\Delta(x,y,z)$, then $\Delta'$ has cone angles $2a,2b,2c$ at these points. Every other point of $\Delta'$ has cone angle at least $2\pi$; this is because all interior points of $\Delta(x,y,z)$ have angle at least $2\pi$ and all non-vertex points on the boundary of $\Delta$ have interior angle at least $\pi$ since the sides are geodesics. The curvature at a point is $2\pi$ minus the cone angle at that point, and each of the $2N$ cone points in $\Delta'$ coming from the $N$ cone points in the interior of $\Delta$ have curvature $\leq -\pi$.  The Gauss--Bonnet theorem implies the total curvature of $\Delta'$ (the sum of the curvatures over all the cone points) is $2\pi \chi(\Delta') = 4\pi$, and therefore
\[4 \pi \le \! (2\pi - 2a) \! + \! (2\pi - 2b) \! + \! (2\pi - 2c) - 2N\pi  <  6\pi - 2N\pi.\]
Thus $2N\pi < 2\pi$, which is only possible if $N = 0$.
\end{proof}

The next lemma will be our main tool for decomposing triangles in the base fiber.

\begin{lemma}\label{L:triangle decomposition}
Let $x,y,z \in \Sigma_0 \subset E_0$ be cone points defining a nondegenerate triangle $\Delta(x,y,z)$.  Let $\gamma$ be the flat geodesic from $x$ to a cone point $w$ in the interior of the opposite side $[y,z]$ in $\Delta(x,y,z)$.  Then $\gamma$ is the union of a proper subsegment $\nu$ of either $[x,y]$ or $[x,z]$ and a single saddle $\mu$ in the interior of $\Delta(x,y,z)$ ending at $w$.

\end{lemma}

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale = .9]
\draw(0,0) -- (2.5,0) -- (1.25,2) -- (0,0);
\draw(3.5,0) -- (6,0) -- (4.75,2) -- (3.5,0);
\draw(7,0) -- (9.5,0) -- (8.25,2) -- (7,0);
\node at (0,-.25) {$y$};
\node at (1.25,2.2) {$x$};
\node at (2.5,-.25) {$z$};
\node at (1.25,-.25) {$w$};
\node at (3.5,-.25) {$y$};
\node at (4.75,2.2) {$x$};
\node at (6,-.25) {$z$};
\node at (5,-.25) {$w$};
\node at (7,-.25) {$y$};
\node at (8.25,2.2) {$x$};
\node at (9.5,-.25) {$z$};
\node at (8,-.25) {$w$};
\draw[ultra thick] (1.25,0) -- (1.25,2);
\draw[ultra thick] (4.75,2) -- (5.375,1) -- (5,0);
\draw[ultra thick] (8.25,2) -- (7.625,1) -- (8,0);
\node at (1.5,.75) {$\mu$};
\node at (5,.5) {$\mu$};
\node at (8.05,.5) {$\mu$};
\node at (5.3,1.6) {$\nu$};
\node at (7.8,1.6) {$\nu$};
\end{tikzpicture}
\caption{Possible configurations of geodesic segments from $x$ to a cone point $w$ in the interior of $[y,z]$ (shown as thickened lines).  The portion in the interior of $\Delta(x,y,z)$ is a single saddle connection.  In the left-most picture $\nu$ is just the point $x$.}
\label{F:opposite sides}
\end{center}
\end{figure}

\begin{proof}
The geodesic $\gamma$ must lie in $\Delta(x,y,z)$ by convexity (Lemma \ref{L:no cones inside}). Starting from $x$, if $\gamma$ does not immediately enter the interior of $\Delta(x,y,z)$, then it must run along one of the edges for some time, then leave that edge. Note that once it leaves the edge it cannot return to it as this would produce two geodesic segments between a pair of points, contradicting the CAT(0) condition.  The segment cannot follow that edge all the way to one of the other vertices, for then it can be continued beyond $w$ to the third vertex, giving two distinct geodesics between $x$ and the third vertex, a contradiction again (because $\Delta(x,y,z)$ is nondegenerate).  It must therefore leave the edge it is following before reaching the vertex.  By a similar reasoning, the segment cannot run along the side opposite $x$ for any of its length, for we could then again continue to another vertex of the triangle and produce two distinct geodesics between a pair of points, a contradiction once again.  Finally, the fact that the part in the interior of $\Delta(x,y,z)$ is a single saddle connection comes from the fact that there are no cone points in the interior by Lemma \ref{L:no cones inside}.
\end{proof}

\subsection{Euclidean triangles}
We now begin by analyzing the simplest type of non-degenerate triangles. Given $x,y,z \in \Sigma_0$, the triangle $\Delta(x,y,z)$ is called {\em Euclidean}, if it is nondegenerate, and each side consists of a single saddle connection.

\begin{lemma} \label{L:triangle balance pt}
For any Euclidean triangle $\Delta(x,y,z)$, there exists a unique point $X \in D$ such that $f_X(\Delta(x,y,z))$ is an equilateral triangle.
\end{lemma}
We call the point $X$ in this lemma the {\em balance point of $\Delta(x,y,z)$}.
\begin{proof} Recall that the points of $D$ parameterize {\em all} affine deformations of the flat metric $q$.  Since $\Delta(x,y,z)$ bounds a $2$--simplex with no cone points in its interior (Lemma~\ref{L:no cones inside}), we may develop the entire $2$--simplex into $\mathbb C$ via preferred coordinates for $q$.  The image is a triangle in $\mathbb C$, and we choose any $A \in \SL_2(\mathbb R)$ so that the associated real linear transformation sends the triangle to an equilateral triangle.  Setting $(X,q_X) = A \cdot (X_0,q)$, we have $f_X(\Delta(x,y,z))$ is equilateral since the preferred coordinates for $A \cdot q$ differ from those for $q$ by composing with $A$; see \S\ref{S:Teich disks defined}.  If $Y \in D$ and $f_Y(\Delta(x,y,z))$ is equilateral, then the affine map $f_{Y,X}$ maps the equilateral triangle $f_X(\Delta(x,y,z))$ to the equilateral triangle $f_Y(\Delta(x,y,z))$.  Therefore $f_{X,Y}$ is conformal, and hence $X = Y$.
\end{proof}

\begin{lemma} \label{L:bounded diameter Euclidean}  There exists $A> 0$ so that the area of any Euclidean triangle $\Delta(x,y,z)$ is at most $A$.  Consequently, if $X$ is the balance point of $\Delta(x,y,z)$, then each side of $f_X(\Delta(x,y,z))$ has length at most $2 \sqrt{A}$. In particular, the balance point lies uniformly close to all 3 horoballs corresponding to the directions of the sides of $\Delta(x,y,z)$.
\end{lemma}
\begin{proof} Let $\Delta(x,y,z)$ be any Euclidean triangle, where $x,y,z\in \Sigma_0$, and let $X$ be its balance point. Since $f_X\colon E_0 \to E_X$ is area-preserving, it suffices to bound the area of the equilateral triangle $\Delta_X = f_X(\Delta(x,y,z))$.
First suppose that $X \in B_\alpha$ for some $\alpha \in \CP$. The flat metric $E_X$ decomposes as a union of strips in the direction $\alpha$, each of which separates $E_X$ and has no cone points in its interior. Thus we may choose adjacent saddle connection edges of $\Delta_X$ that both traverse the interior of the same strip $W$ and meet, with an angle of $\pi/3$, at the boundary of $W$; see Figure~\ref{F:euclidean crossing strip}.
The intersection of $\Delta_X$ with the opposite boundary of $W$ then has length at least $2w/\sqrt{3}$, where $w$ is the width of the strip $W$. On the other hand, since $\Delta_X$ contains no cone points in its interior (Lemma~\ref{L:no cones inside}), it intersects at most one saddle connection on the opposite boundary of $W$. As this saddle connection has bounded length, since it has direction $\alpha$ and $X\in B_\alpha$ (see Proposition~\ref{P:horoball sublevel set} and Lemma~\ref{L:strip-and-saddle-bound}), we obtain a uniform bound on the width $w$ of $W$. Therefore $X$ can be only uniformly deep in $B_\alpha$.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale = 1.5]
\shadedraw[line width=0] (.8,-.35) -- (1,.5) -- (2.1,-.1);
\draw[fill, opacity = .05] (-.5,.1) -- (3.5,.1) --  (3.5,.5) -- (-.5,.5) -- (-.5,.1);
\draw[line width = .5pt] (0,.1) -- (3,.1);
\draw[line width = .5pt, dotted] (-.5,.1) -- (0,.1);
\draw[line width = .5pt, dotted] (3,.1) -- (3.5,.1);
\draw[fill]  (.1,.1) circle (.03);
\draw[fill]  (.8,.1) circle (.03);
\draw[fill]  (2,.1) circle (.03);
\draw[fill]  (2.7,.1) circle (.03);
\draw[line width = .5pt] (0,.5) -- (3,.5);
\draw[line width = .5pt, dotted] (-.5,.5) -- (0,.5);
\draw[line width = .5pt, dotted] (3,.5) -- (3.5,.5);
\draw[fill]  (.5,.5) circle (.03);
\draw[fill]  (1,.5) circle (.03);
\draw[fill]  (1.5,.5) circle (.03);
\draw[fill]  (2.6,.5)  circle (.03);
\draw[line width = 1pt] (.8,-.35) -- (1,.5) -- (2.1,-.1);
\node at (0,.3) {$W$};
\node at (1.35,-.05) {$\Delta_X$};
\end{tikzpicture}
\caption{Adjacent sides of $\Delta_X$ traversing a strip $W$} \label{F:euclidean crossing strip}
\end{center}
\end{figure}

The argument above proves that $X$ is in some uniformly thick part of $D$. Thus there is a uniform bound $r$ on the diameter of $E_X/\pi_1S$. Then the diameter of $\Delta_X$ must be at most $4r$, as otherwise it would contain a ball of radius $r$ in its interior, and this would cover $E_X/\pi_1S$, proving that the image contains no cone points, a contradiction. Therefore the equilateral triangle $\Delta_X$ has area at most $4r^2\sqrt{3}$.
\end{proof}

\subsection{Fans}
The next simplest type of nondegenerate triangles are those built from Euclidean triangles in the pattern of a ``fan".  The analysis of the associated collapsed preferred paths of triangles in this case is the key technical result behind the proof of Theorem~\ref{T:preferred slim triangles}.  We now proceed to the precise description of these triangles and their analysis.

Given $x,y,z\in \Sigma$, the triangle $\Delta(x,y,z)$ in $E_0$ is called a {\em fan} if  it is nondegenerate and at least two sides consist of a single saddle connection.  By Lemma \ref{L:triangle decomposition}, each geodesic between a cone point in the interior of the third side to the vertex opposite that side is a single saddle connection.  It follows that fans decompose into unions of Euclidean triangles, all of which have a common vertex.  See Figure~\ref{F:fan}.

We say that two saddle connections in a fiber $E_X$ \emph{span a triangle} if they share an endpoint and the flat geodesic joining their other endpoints is a single (possibly degenerate) saddle connection.  Given a saddle connection $\sdl$ in some fiber $E_X$, let us write $\mathcal{P}(\sdl)\subset \mathcal{P}$ for the set of directions of all saddle connections that span a triangle with $\sdl$. Notice that the direction of $\sdl$ is contained in $\mathcal P(\sdl)$ since $\sdl$ spans a (degenerate) triangle with itself. Denote
\[B(\sdl) = \bigcup_{\alpha\in \mathcal{P}(\sdl)} B_{\alpha}\]
for the union of horoballs at these directions. Similarly, taking horoball preimages, we denote $\mathcal{B}(\sdl) =\bigcup_{\alpha\in \mathcal{P}(\sdl)} \mathcal B_{\alpha}$.

The utility of this notion comes from the following corollary of Lemma \ref{L:bounded diameter Euclidean}.

\begin{corollary}\label{cor:balance_point_near_horob}
 There exists $A'>0$ so that for each saddle connection $\sdl$ in the direction $\alpha$ the following holds. For any $\beta\in \mathcal P(\sdl)$ we have $d(B_\alpha,B_\beta)<A'$ in $D$.
\end{corollary}

We now show that fans give rise to slim triangles of collapsed preferred paths, and we moreover give a criterion, in terms of the notions that we just defined, for the triangle to be coarsely degenerate.

\begin{lemma}[Fan lemma]
\label{L:slim_fans}
There exists a constant $\delta' > 0$ such that if the geodesic triangle $\Delta(x,y,z)$ in $E_0$ is a fan, where $x,y,z\in \Sigma$, then the triangle $\Delta^{\hat\varsigma}(x,y,z)$ of collapsed preferred paths is $\delta'$--slim. If, furthermore, $[f(x),f(y)]=\sdl_z$ and $[f(y),f(z)]=\sdl_x$ are each a single saddle connection and $y$ lies on a geodesic in $D_y$ with respective endpoints in $\mathcal B(\sdl_z)$ and $\mathcal B(\sdl_x)$, then the collapsed preferred paths satisfy
\[\hat\varsigma(x,y), \hat\varsigma(y,z) \subset N_{\delta'}\left(\hat\varsigma(x,z)\right)
\qquad\text{and}\qquad
\hat\varsigma(x,z)\subset N_{\delta'}(\hat\varsigma(x,y)\cup \hat\varsigma(y,z)).\]
\end{lemma}

We will make frequent (sometimes implicit) use of the following fact about the geometry of the hyperbolic plane; see e.g. \cite[Lemma 4.5]{GM08}.
\begin{lemma} \label{L:geodesics between horoballs} If points $v,w\in D$ respectively lie within bounded distance of horoballs $B_\alpha,B_\beta$, for $\alpha,\beta\in \mathcal{P}$, then the geodesic joining $v$ to $w$ lies within a uniform neighborhood of $B_\alpha\cup B_\beta$ and the shortest geodesic joining these horoballs. \qed
\end{lemma}
Hence, if $h_1, h_2$ are horizontal geodesics in some Teichm\"uller disc $D_x$ whose endpoints lie within bounded distance of the same horoball preimages $\mathcal{B}_\alpha$ and $\mathcal{B}_\beta$, then $P(h_1)$ and $P(h_2)$ have bounded Hausdorff distance in $\hat E$.

\begin{proof}[Proof of Lemma \ref{L:slim_fans}] In what follows, we will often lift objects (points, paths, etc) from $D$ to a horizontal disk $D_w$, through a point $w \in E$.  When we can do so without confusion, we will use the same name for the object in $D$ and in $D_w$, to avoid introducing even more notation than is already necessary.

As depicted in Figure~\ref{F:fan}, we can express the three sides of $\Delta(x,y,z)\subset E_0$ as concatenations of saddle connections:
\[[f(x),f(y)] = \sdl_z, \quad [f(y),f(z)] = \sdl_x, \quad\text{and}\quad [f(x),f(z)] = \sdl_1\sdl_2\dotsb\sdl_k.\]
For each index $\ast\in\{z,x,1,\dotsc,k\}$, write $\alpha_*\in\mathcal{P}$ for the direction of the saddle connection $\sdl_*$ and set $\gamma_* = f_{X_{\alpha_*}}(\sdl_*)$. We may then write
\[\varsigma(x,y) = h_z \gamma_z h'_z \quad\text{and}\quad
\varsigma(y,z) = h'_x \gamma_x h_x,\]
where $h_z$ is the horizontal geodesic from $x$ to $X_{\alpha_z}$ in $D_x$, $h'_z$ the geodesic from $X_{\alpha_z}$ to $y$ in $D_y$, $h'_x$ the geodesic from $y$ to $X_{\alpha_x}$ in $D_y$, and $h_x$ the geodesic from $X_{\alpha_x}$ to $z$ in $D_z$. We also have
\[\varsigma(x,z) = h_0 \gamma_1 h_1 \gamma_2 h_2 \dotsb \gamma_k h_k\]
where $h_0$ is is the geodesic from $x$ to $X_{\alpha_1}$ in $D_x$, $h_k$ the geodesic from $X_{\alpha_k}$ to $z$ in $D_z$, and $h_i$ the geodesic from $X_{\alpha_i}$ to $X_{\alpha_{i+1}}$ in $D_{\gamma_i^+} = D_{\gamma^-_{i+1}}$ for each $1\le i < k$.

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale = 1.2]
\usetikzlibrary{calc,patterns,angles,quotes}
\coordinate (y) at (0,3);
\coordinate (x) at (-3.5,-1);
\coordinate (cone) at (-1.9,-.3);
\coordinate (ctwo) at (-.8,-.1);
\coordinate (cthree) at (.7, -.1);
\coordinate (cfour) at (1.7,-.2);
\coordinate (z) at (3.5,-.75);
\coordinate (T_i) at (0, 1);
\coordinate (tau-) at (-.8, .75);
\coordinate (tau+) at (.8, .75);
\node[above] at (y) {$f(y)$};
\node[left] at (x) {$f(x)$};
\node[right] at (z) {$f(z)$};
\node at (T_i) {$T_i$};
\node at (tau-) {$\tau_i^-$};
\node at (tau+) {$\tau_i^+$};
\draw (y) -- node[left] {$\sdl_z$} (x);
\draw (y) -- node[right] {$\sdl_x$} (z);
\draw (y) -- (cone);
\draw (y) -- (ctwo);
\draw (y) -- (cthree);
\draw (y) -- (cfour);
\draw (x) --node[below] {$\sdl_1$}  (cone) --  (ctwo) --node[below] {$\sdl_i$}  (cthree) --  (cfour) --node[below]{$\sdl_k$} (z);

\pic [draw, "$\lambda_1$", angle eccentricity=2] {angle = cone--x--y};
\pic [draw, "$\rho_1$"] {angle = y--cone--x};

\pic [draw, "$\lambda_i$"] {angle = cthree--ctwo--y};
\pic [draw, "$\rho_i$"] {angle = y--cthree--ctwo};

\pic [draw, "$\lambda_k$"] {angle = z--cfour--y};
\pic [draw, "$\rho_{k}$", angle eccentricity=2] {angle = y--z--cfour};

\end{tikzpicture}
\caption{A fan $\Delta(x,y,z) \subset E_0$, in which $[f(x),f(y)]$ and $[f(y),f(z)]$ are both a single saddle connection.}
\label{F:fan}
\end{center}

\end{figure}

Let $h'_y$ be the horizontal geodesic from $X_{\alpha_z}$ to $X_{\alpha_x}$ in $D_y$. The three paths $h'_z$, $h'_y$, $h'_x$ form a geodesic triangle in the $2$--hyperbolic space $D_y$; thus we have
\[
h'_z \subset N_{2}(h'_x\cup h'_y),\quad
h'_x \subset N_{2}(h'_z\cup h'_y),\quad\text{and}\quad
h'_y \subset N_{2}(h'_z\cup h'_x).
\]
Since $P\colon E\to \hat{E}$ is $1$--Lipschitz; the collapsed horizontal segments $P(h'_z)$, $P(h'_x)$, and $P(h'_y)$ also form a $2$--slim triangle. Therefore, the first conclusion of the lemma follows from the following claim, which we prove below.

\begin{claim}
\label{claim:truncated_fan}
The collapsed paths $P(h_z \gamma_z h'_y \gamma_x h_x)$ and $P(h_0\gamma_1 h_1 \gamma_2 h_2 \dotsc \gamma_k h_k) = P(\varsigma(x,z))$ have uniformly bounded Hausdorff distance.
\end{claim}

Now we argue that the claim also implies the `furthermore' conclusion of the lemma.
Assume that $y$ lies on a geodesic $g$ in $D_y$ with endpoints in $\mathcal B(\sdl_z)$ and $\mathcal B(\sdl_x)$. This means that there are saddle connections $\sdl'_z$ and $\sdl'_x$ spanning triangles $T_z$ and $T_x$ with $\sdl_z$ and $\sdl_x$, respectively, such that the endpoints of $g$ lie over the horoballs $B'_z$ and $B'_x$ for the directions of $\sdl'_z$ and $\sdl'_x$ (meaning that $\pi$ of the endpoints lie on the prescribed horoballs).
By Corollary \ref{cor:balance_point_near_horob}, the endpoints of $h'_y$ lie over horoballs close to $B'_z$ and $B'_x$, respectively. In particular, $P(g)$ and $P(h'_y)$ lie within uniformly bounded Hausdorff distance (Lemma~\ref{L:geodesics between horoballs}).

We now see that the sets
\[P(h'_z)\cup P(h'_x)\quad\text{and}\quad P(h'_y)\]
have uniformly bounded Hausdorff distance, since $h'_z$ is a geodesic sharing an endpoint with $h'_y$, while the $P$--image of the other endpoint is close to $P(h'_y)$, and similarly for $h'_x$. Therefore, the `furthermore' conclusion of the lemma follows from Claim~\ref{claim:truncated_fan} as well.
\end{proof}

To prove Claim~\ref{claim:truncated_fan}, let $T_i\subset E_0$ be the Euclidean triangle determined by $f(y)$ and the saddle connection $\sdl_i$, that is, with vertices $f(y)$, $\sdl_i^-$, and $\sdl_i^+$. We call $\sdl_i$ the base of $T_i$ and $\tau_i^\pm = [\sdl_i^\pm,f(y)]$ the sides of $T_i$. Let $\lambda_i$ and $\rho_i$ denote the angles of $T_i$ between $\sdl_i$ and the sides $\tau_i^-$ and $\tau_i^+$, respectively; see Figure~\ref{F:fan}. For any $X\in D$, we write $T_i(X)$, $\sdl_i(X)$, $\lambda_i(X)$, etc., for the corresponding pieces of the image fan $\Delta_X = f_X(\Delta(x,y,z))$ in the fiber $E_X$. Let $b_i\in D$ denote the balance point for the Euclidean triangle $T_i$, and $h$ the geodesic in $D$ from $b_1$ to $b_k$.

The next claim is the key to proving Claim~\ref{claim:truncated_fan} (and is perhaps the biggest miracles in the proof of hyperbolicity of $\hat E$).  Roughly, it says that as we traverse $h$ from $b_1$ to $b_k$ in $D$, each of the saddle connections $\sdl_1,\ldots,\sdl_k$ become bounded in length when viewed in the fibers over $h$, and that the times when they are bounded occur in order.

\begin{claim}
\label{claim:fan_hoiz_balancpts}
There are points $t_1,\dotsc, t_k$ appearing in order along the geodesic $h$ that respectively lie within uniformly bounded distance of the horoballs $B_{\alpha_1}, \dotsc B_{\alpha_k}$.
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:fan_hoiz_balancpts}]
We find points $t_i$ as in the statement with $\mathrm{len}(\sdl_i(t_i))$ uniformly  bounded for each $i=1,\dotsc, k$. By Lemma~\ref{L:bounded diameter Euclidean}, we may take $t_1 = b_1$ and $t_k = b_k$.

First observe that for any $X\in D$ we have
\[\lambda_1(X) < \lambda_2(X) < \dotsc < \lambda_k(X)
\quad\text{and}\quad
\rho_1(X) > \rho_2(X) > \dotsb > \rho_k(X).\]
Indeed, $\lambda_i(X) + \rho_i(X) < \pi$ by the fact that these are two angles of the Euclidean triangle $T_i(X)$, whereas $\rho_i(X) + \lambda_{i+1}(X) \ge \pi$ by the fact that $\sdl_1(X)\dotsb\sdl_k(X)$ is a geodesic in $E_X$. Thus $\lambda_i(X) < \lambda_{i+1}(X)$, proving the claimed inequalities for the $\lambda_i$; the ones for the $\rho_i$ can be proven similarly.

Since $T_1(b_1)$ and $T_k(b_k)$ are equilateral, for each $1  < i < k$ we thus have
\[\rho_i(b_1) < \rho_1(b_1) = \frac{\pi}{3} = \lambda_k(b_k) > \lambda_i(b_k).\]
On the other hand, since $\sdl_1(X)\dotsb\sdl_k(X)$ forms a geodesic in any fiber $E_X$, we also have that $\rho_{k-1}(b_k) + \lambda_k(b_k) \ge \pi$ and $\rho_1(b_1)+\lambda_2(b_1) \ge \pi$. Hence, again for each $1  < i < k$, we have
\[
\rho_i(b_k) \ge \rho_{k-1}(b_k) \ge \frac{2\pi}{3}
\le \lambda_2(b_1) \le \lambda_i(b_1).
\]
Using the intermediate value theorem, for each $1 < i < k$ we  define $l_i,r_i\in D$ to be the first points along the geodesic $h$ such that $\lambda_i(l_i) = \pi/2$ and $\rho_i(r_i) = \pi/2$.

Let us use $\le$ to denote the natural order along $h$ (oriented from $b_1$ to $b_k$). Notice that $\lambda_i(r_i) < \pi/2$ (since $\rho_i(r_i) + \lambda_i(r_i) < \pi$). As we also have $\lambda_i(b_1) > \pi/2$, the definition of $l_i$ implies that $l_i \le r_i$. The fact $\rho_i(l_{i+1}) + \lambda_{i+1}(l_{i+1}) \ge \pi$ further implies $\rho_i(l_{i+1}) \ge \pi/2$. Since $\rho_i(b_1) < \pi/2$, the definition of $r_i$ gives $r_i \le l_{i+1}$. Thus our points along the geodesic $h=[b_1, b_k]$ are ordered as
\[b_1 = t_1 \le l_2 \le r_2 \le \dotsc \le l_{k-1} \le r_{k-1} \le t_k = b_k.\]

To complete the proof, it now suffices to show, for each $1 < i < k$, that there is a point $t_i\in [l_i,r_i]$ with $\sdl_i(t_i)$ of uniformly bounded length. By Lemma~\ref{L:bounded diameter Euclidean}, we have
\begin{align*}
\mathrm{len}(\tau_i^-(l_i))\cdot\mathrm{len}(\sdl_i(l_i)) &=2\mathrm{area}(T_i(l_i)) \le 2A,\quad\text{and}\\
\mathrm{len}(\tau_i^+(r_i))\cdot\mathrm{len}(\sdl_i(r_i)) &= 2\mathrm{area}(T_i(r_i)) \le 2A.
\end{align*}
If $\mathrm{len}(\sdl_i(l_i)) \le 2\sqrt{A}$ or $\mathrm{len}(\sdl_i(r_i)) \le 2\sqrt{A}$ we are done. Otherwise  $\mathrm{len}(\tau_i^-(l_i))\le \sqrt{A}$ and $\mathrm{len}(\tau_i^+(r_i))\le \sqrt{A}$. Thus $l_i$ lies within bounded distance of the horoball for the direction $\tau_i^-$, and $r_i$ lies within bounded distance of the horoball for the direction $\tau_i^+$. Since the balance point $b_i$ for $T_i$ lies lies close to both of these horoballs by Lemma \ref{L:bounded diameter Euclidean}, the balance point lies near the shortest geodesic joining the horoballs.  It follows from Lemma \ref{L:geodesics between horoballs} that the geodesic $[l_i,r_i]$ passes near $b_i$, at which point $\sdl_i$ has bounded length, again by Lemma~\ref{L:bounded diameter Euclidean}.
\end{proof}

We now commence with the proof of Claim~\ref{claim:truncated_fan}.

\begin{proof}[Proof of Claim~\ref{claim:truncated_fan}]
Let $h_y$ denote the horizontal geodesic in $D_y$ from $b_1$ to $b_k$, that is, $h_y$ is the horizontal lift of $h$ to $D_y$. By Lemma~\ref{L:bounded diameter Euclidean}, $b_1$ and $b_k$ respectively lie within bounded distance of the horoballs $B_z$ and $B_x$. Since $h'_y$ runs between these horoballs as well, we see that $P(h'_y)$ and $P(h_y)$ have bounded Hausdorff distance.
The horizontal paths $h_z$ and $h_0$ in $D_x$ go from $x$ to the respective horoballs $B_{\alpha_z}$ and $B_{\alpha_1}$ which lie bounded distance from each other (they are near the balance point $b_1$ for $T_1$). Thus we similarly conclude that $P(h_z)$ and $P(h_0)$ have bounded Hausdorff distance. Symmetrically, the same holds for $P(h_x)$ and $P(h_k)$.

By the above remarks, and using the facts that $P$ is Lipschitz and the vertical paths $\gamma_z, \gamma_x, \gamma_1,\dotsc, \gamma_k$ each have uniformly bounded length once projected to $\hat E$ by $P$, it suffices to show that the sets
\[P(h_y)\quad\text{and}\quad P(h_1 \cup \dotsb \cup h_{k-1})\]
have bounded Hausdorff distance.

For $1 \le i < k$, let $g_i$ denote the horizontal geodesic in $D_y$ between the balance points $b_i$ and $b_{i+1}$ of the triangles $T_i$ and $T_{i+1}$. Similarly, let $h_i'$ be the horizontal geodesic between these points $b_i$ and $b_{i+1}$ in the disc $D_{\gamma_i^+} = D_{\gamma_{i+1}^-}$. Since the saddle connection $\tau_{i}^+ = \tau_{i+1}^-$ from $\gamma_i^+= \gamma_{i+1}^-$ to $f(y)$ is common to both triangles $T_i$ and $T_{i+1}$, Lemma~\ref{L:bounded diameter Euclidean} implies that $b_i$ and $b_{i+1}$ both lie within bounded distance of the horoball for the direction of $\tau_{i}^+$. Thus the saddle connection $\tau_i^+$ has bounded length over the whole geodesic $[b_i,b_{i+1}]$, and we conclude that the paths $g_i$ and $h_i'$ have bounded Hausdorff distance in $E$.

Using Claim~\ref{claim:fan_hoiz_balancpts}, let $s_i$ denote the lift of $t_i\in h$ to $h_y$. This gives a decomposition $h_y = h_y^1\dotsb h_y^{k-1}$ of $h_y$ into segments
\[h_y^i = [s_i, s_{i+1}],\quad\text{for $i = 1,\dotsc, k-1$}.\]
By construction of $t_i$, we see that $h_y^i$ begins and ends near the horoballs $B_{\alpha_i}$ and $B_{\alpha_{i+1}}$ in $D_{\gamma_i^+}$.  The balance points $b_i$ and $b_{i+1}$ lie near these horoballs as well. Since $P(h^i_y)$ and $P(h'_i)$ are at bounded Hausdorff distance, it follows that $P(h_y^i)$ and $P(g_i)$ are at bounded Hausdorff distance. Combining with the previous paragraph, this implies $P(h_y)$ and $P(h_1'\cup\dotsb h_{k-1}')$ have bounded Hausdorff distance.

To complete the proof of Claim~\ref{claim:truncated_fan}, it now suffices to show that
\[P(h'_i)\quad\text{and}\quad P(h_i)\]
have uniformly bounded Hausdorff distance for each $1 \le   i < k$. But this is clear: The endpoints $b_i$ and $b_{i+1}$ of $h'_i$ lie uniformly close to the horoballs $B_{\alpha_i}$ and $B_{\alpha_{i+1}}$ containing the endpoints $X_{\alpha_i}$ and $X_{\alpha_{i+1}}$ of $h_i$. Since both paths $h'_i$ and $h_i$ lie in the same disc $D_{\gamma_i^+} = D_{\gamma_{i+1}^-}$, it follows that $P(h'_i)$ and $P(h_i)$ have uniformly bounded Hausdorff distance. This establishes Claim~\ref{claim:truncated_fan} and concludes the proof of Lemma~\ref{L:slim_fans}.
\end{proof}

\subsection{Decomposing triangles into fans}

We next consider the case that $\Delta(x,y,z)$ is a triangle where one side is a single saddle connection.  The main idea is that such triangles decompose into an alternating union of fans at certain ``pivot'' cone points along the sides with more than one saddle connection, as in Figure \ref{F:fans_and_pivots}.

\begin{lemma} \label{L:length one side slim} There exists $\delta''> 0$ so that if $x,y,z \in \Sigma$ and $\Delta(x,y,z)$ is a nondegenerate triangle so that one side is a saddle connection, then $\Delta^{\hat \varsigma}(x,y,z)$ is $\delta''$--slim.
\end{lemma}
\begin{proof} Let $\delta' > 0$ be the constant provided by Lemma~\ref{L:slim_fans}. We may assume that the side $[f(x),f(y)]$ consists of a single saddle, which we denote $\eta$. To economize notation, set $x' =y$. Write $[f(x),f(z)] = \sdl_1\dotsb\sdl_m$ as a concatenation of saddle connections, and let $f(x) = a_0,a_1,\dotsc,a_m=f(z)$ be the sequence of endpoints of these saddle connections. Similarly write $[f(x'),f(z)] = \sdl'_1\dotsb\sdl'_n$ with $f(x') = a'_0,a'_1,\dotsc,a'_n=f(z)$ the corresponding sequence of cone points.

Letting $\alpha,\alpha_i,\alpha'_j\in\CP$ be the directions of $\eta$, $\sdl_i$ and $\sdl'_j$, respectively, we then have
\begin{equation}
\label{eqn:some_preferred_paths}
\varsigma(x,z) = h_0\gamma_1 h_1\gamma_2\dotsb \gamma_m h_m
\qquad\text{and}\qquad
\varsigma(x',z)=h'_0\gamma'_1 h'_1 \gamma'_2\dotsb \gamma'_n h'_n,
\end{equation}
where $\gamma_i = f_{X_{\alpha_i}}(\sdl_i)$ and each $h_i\subset D_{a_i}$ is a horizontal geodesic making the concatenation into a path, and similarly for $\gamma'_j = f_{X_{\alpha'_j}}(\sdl'_j)$ and $h'_j \subset D_{a'_j}$.

We now explain how to decompose $\Delta(x,y,z)$ into a union of fans as in Figure \ref{F:fans_and_pivots}, where the base points (called pivots) of the fans alternate sides of the triangle.

First, we claim that there is a single saddle connection joining $f(x) =a_0$ to $a'_1$ or else there is a single saddle connection joining $f(x')=a'_0$ to $a_1$ (or both).  If not, then the geodesics $[a_0,a'_1]$ and $[a'_0,a_1]$ are both nontrivial concatenations of saddle connections joining cone points in $\Delta(x,x',z)\subset E_0$.

Since $[a_0, a'_0]$ is a single saddle connection, Lemma \ref{L:triangle decomposition} implies that $\sdl_1 = [a_0,a_1]$ is the first saddle connection in the geodesic $[a_0,a'_1]$.  In particular, $[a_1,a'_1]$ is a subpath of $[a_0,a'_1]$. Similarly, $\sdl'_1=[a'_0,a'_1]$ is the first saddle connection of the geodesic $[a'_0,a_1]$, which contains $[a'_1,a_1]$ as a subpath. These observations imply that $\sdl_1[a_1,a'_1]\overline{\sdl'_1}$ is a concatenation of saddle connections making angle at least $\pi$ on both sides of each cone point encountered. Therefore it is the geodesic in $E_0$ connecting $f(x)$ to $f(x')$. But we are assuming that this geodesic consists of a single saddle connection, a contradiction.

By the previous paragraph, we may without loss of generality assume that $[a_0,a'_1]$ is a single saddle connection. Let $j_1\in \{1,\dotsc,n\}$ be the largest index such that $[a_0,a'_{j_1}]$ is a single saddle connection.

If $j_1 = n$, then $\Delta(x,x',z)$ is evidently a fan and the conclusion follows from Lemma~\ref{L:slim_fans}. We may therefore assume $j_1 < n$, in which case the cone points $a_0,a'_{j_1},f(z)$ span a nondegenerate geodesic triangle with the side $[a_0,a'_{j_1}]$ consisting of a single saddle connection. The above observations now imply that $[a'_{j_1},a_1]$ is a single saddle connection, and we are justified in letting $i_1\in \{1,\dotsc, m\}$ be the largest index such that $[a'_{j_1},a_{i_1}]$ is a single saddle connection.

Continuing in this manner, we recursively choose indices $j_1,i_1,j_2,i_2,\dotsc$, terminating when some $j_k = n$ or some $i_k = m$, with  the defining property that $j_{k+1}$ is the largest index in $\{j_{k}+1,\dotsc, n\}$ such that $[a_{i_k},a'_{j_{k+1}}]$ is a single saddle connection and $i_{k+1}$ is the largest index in $\{i_{k}+1, \dotsc, m\}$ such that $[a'_{j_{k+1}},a_{i_{k+1}}]$ is a single saddle connection. In this way, we decompose the triangle $\Delta(x,x',z)$ into fans based at the {\em pivot vertices} $a_0,a'_{j_1},a_{i_1},\dotsc$. For the sake of argument, we may assume some $i_k = n$ so that the situation is as depicted in Figure~\ref{F:fans_and_pivots}.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale = 1]
\draw(0,0) -- (-10,-2) -- (-10,2) -- (0,0);
\draw[line width=1.5] (0,0) -- (-2,-.4);
\draw (-2,-.4) -- (-.5,.1);
\draw (-2,-.4) -- (-1,.2);
\draw (-2,-.4) -- (-1.5,.3);
\draw (-2,-.4) -- (-2,.4);
\draw (-2,-.4) -- (-2.5,.5);
\node at (-2,-.4) [below]  {$a'_{j_4}$};
\draw[line width=1.5] (-2,-.4) -- (-3,.6);
\node at (-3, .6) [above] {$a_{i_3}$};
\draw[line width=1.5] (-3,.6) -- (-3,-.6);
\draw (-3,-.6) -- (-3.5,.7);
\draw (-3,-.6) -- (-4,.8);
\draw (-3,-.6) -- (-4.5,.9);
\node at (-3,-.6) [below] {$a'_{j_3}$};
\draw[line width=1.5] (-3,-.6) -- (-5,1);
\draw (-5,1) -- (-3.25,-.65);
\draw (-5,1) -- (-3.5,-.7);
\draw (-5,1) -- (-3.75,-.75);
\draw (-5,1) -- (-4,-.8);
\draw (-5,1) -- (-4.25,-.85);
\draw (-5,1) -- (-4.5,-.9);
\draw (-5,1) -- (-4.75,-.95);
\node at  (-5,1) [above] {$a_{i_2}$};
\draw[line width=1.5] (-5,1) -- (-5,-1);
\node at (-5,-1) [below] {$a'_{j_2}$};
\draw[line width=1.5] (-5,-1) -- (-7,1.4);
\draw (-7,1.4) -- (-6,-1.2);
\draw (-7,1.4) -- (-7,-1.4);
\node at (-7,1.4) [above] {$a_{i_1}$};
\draw[line width=1.5] (-7,1.4) -- (-8,-1.6);
\draw (-8,-1.6) -- (-7.5,1.5);
\draw (-8,-1.6) -- (-8,1.6);
\draw (-8,-1.6) -- (-8.5,1.7);
\draw (-8,-1.6) -- (-9,1.8);
\draw (-8,-1.6) -- (-9.5,1.9);
\node at (-8,-1.6) [below] {$a'_{j_1}$};
\draw[line width=1.5] (-8,-1.6) -- (-10,2);
\draw (-10,2) -- (-9.5,-1.9);
\draw (-10,2) -- (-9,-1.8);
\draw (-10,2) -- (-8.5,-1.7);
\draw[line width=1.5] (-10,2) -- (-10,-2);
\node at (-10.4,2.2)  {$f(x)=a_0 = f(x_0)$};
\node at (-10.4,-2.2) {$f(x')=a'_0$};
\node at (.3,0) [above]  {$a_{i_k} = f(z)$};
\end{tikzpicture}
\caption{Decomposing a triangle into fans based at pivot vertices.}
\label{F:fans_and_pivots}
\end{center}
\end{figure}

For each of the pivot vertices, $a_0,a'_{j_1},a_{i_1},\dotsc, a_{i_{k-1}},a'_{j_k}$, we choose a point in the corresponding Teichm\"uller disk as follows: Recall our notation (\ref{eqn:some_preferred_paths}) for the preferred paths $\varsigma(x,z)$ and $\varsigma(x',z)$. Let $x_0\in D_{a_0}$ be the terminal endpoint of the initial horizontal segment $h_0$ of $\varsigma(x,z)$. That is, $x_0$ is the intersection of $h_0$ and $\gamma_1$, and thus lies in the designated fiber $E_{X_{\alpha_1}}$ over the the boundary of the horoball $\partial B_{\alpha_1}$ for the direction $\alpha_1$ of the saddle connection $\sdl_1$.
For each $s \in \{1,\dotsc, k\}$, let $x'_s \in D_{\alpha_{j_s'}}$ be any point on the horizontal segment $h'_{j_s}$ of $\varsigma(x',z)$ corresponding to the vertex $a'_{j_s}$. Similarly, for any $r\in \{1,\dotsc, k-1\}$, let $x_r\in D_{\alpha_{i_r}}$ be any point on the horizontal segment $h_{i_r}$ of $\varsigma(x,z)$ corresponding to the vertex $a_{i_r}$. To round out the notation, we also set $x_{k} = z$. Notice that these choices decompose the preferred paths $\varsigma(x,z)$ and $\varsigma(x',z)$ into concatenations of preferred paths:
\begin{align*}
\varsigma(x,z) &= \varsigma(x,x_0)\varsigma(x_0,x_1)\varsigma(x_1,x_2)\dotsb\varsigma(x_{k-1},z),\text{ and}\\
\varsigma(x',z) &= \varsigma(x',x'_1)\varsigma(x'_1,x'_2)\dotsb\varsigma(x'_k,z).
\end{align*}

Now, for each $r\in \{1,\dotsc, k-1\}$, the three points $x'_r,x_r,x'_{r+1}\in \Sigma$ satisfy all of the hypotheses of the Fan Lemma~\ref{L:slim_fans}, including the furthermore hypothesis on the location of the pivot vertex with respect to the horoballs for the adjacent saddle connections. Therefore, Lemma~\ref{L:slim_fans} implies the sets
\[\hat\varsigma(x'_r, x_r) \cup \hat\varsigma(x_r, x'_{r+1})\quad\text{and}\quad \hat\varsigma(x'_r,x'_{r+1})\]
have Hausdorff distance at most $\delta'$.
Similarly, for each $s\in\{1,\dotsc,k\}$, we may apply Lemma~\ref{L:slim_fans} to the points $x_{s-1},x'_s,x_s\in \Sigma$ to bound the Hausdorff distance between
\[\hat\varsigma(x_{s-1},x'_s) \cup \hat\varsigma(x'_s,x_s) \quad\text{and}\quad \hat\varsigma(x_{s-1},x_s)\]
by $\delta'$. The same reasoning applied to $x',x_0,x'_1$ gives that
\[\hat\varsigma(x',x_0)\cup \hat\varsigma(x_0,x'_1)\quad\text{and}\quad  \hat\varsigma(x',x'_1)\]
have Hausdorff distance at most $\delta'$ as well. Furthermore, the triangle $\Delta^\varsigma(x,x',x_0)$ is $2$--slim, whence so is $\Delta^{\hat\varsigma}(x,x',x_0)$, since $x$ and $x_0$ lie in the same Teichm\"uller disc (after removing the common initial segments of $\varsigma(x',x)$ and $\varsigma(x',x_0)$ from $\Delta^\varsigma(x',x,x_0)$ one is left with a geodesic triangle in the $2$--hyperbolic space $D_{a_0}$).

We now show that the triangle  $\Delta^{\hat\varsigma}(x,x',z)$ is $(2\delta'+2)$--slim. The above shows that the projected path $\hat\varsigma(x',z) = \hat\varsigma(x',x'_1)\cup\dotsb\cup\hat\varsigma(x'_k,z)$ is contained in the $\delta'$--neighborhood of the set
\[\Big(\hat\varsigma(x',x_0)\Big)\cup \Big(\hat\varsigma(x_0,x'_1)\cup\hat\varsigma(x'_1,x_1)\cup\dotsc\cup\hat\varsigma(x_{k-1},x'_k)\cup\hat\varsigma(x'_k,z)\Big),\]
which is in turn contained in the $2+\delta'$ neighborhood of
\[\Big(\hat\varsigma(x',x)\cup \hat\varsigma(x,x_0)\Big)\cup\Big(\hat\varsigma(x_0,x_1)\cup\dotsb\cup\hat\varsigma(x_{k-1},z)\Big) = \hat\varsigma(x',x)\cup\hat\varsigma(x,z).\]

On the other hand, the above also shows that the projected preferred path
\[\hat\varsigma(x,z) = \Big(\hat\varsigma(x,x_0)\Big)\cup\Big(\hat\varsigma(x_0,x_1)\cup\dotsb\cup\hat\varsigma(x_{k-1},z)\Big)\]
is contained in the $(2+\delta')$--neighborhood of
\begin{align*}
\Big(\hat\varsigma(x,x')&\cup\hat\varsigma(x',x_0)\Big)\cup\Big(\hat\varsigma(x_0,x'_1)\cup\hat\varsigma(x'_1,x_1)\cup\dotsb\cup\hat\varsigma(x_{k-1},x'_k)\cup \hat\varsigma(x'_k,z)\Big)\\
=&\hat\varsigma(x,x')\cup\Big(\hat\varsigma(x',x_0)\cup\hat\varsigma(x_0,x'_1)\cup\hat\varsigma(x'_1,x_1)\cup\dotsb\cup\hat\varsigma(x_{k-1},x'_k)\cup \hat\varsigma(x'_k,z)\Big)
\end{align*}
which is in turn contained in the $\delta'$--neighborhood of
\[\hat\varsigma(x,x')\cup \Big(\hat\varsigma(x',x'_1)\cup\hat\varsigma(x'_1,x'_2)\cup\dotsb\cup\hat\varsigma(x'_k,z)\Big) = \hat\varsigma(x,x')\cup \hat\varsigma(x',z).\]

Finally, we see that $\hat\varsigma(x,x')$ is contained in the $2$--neighborhood of
\[\hat\varsigma(x,x_0)\cup \hat\varsigma(x_0,x')\]
which itself is contained in the $\delta'$--neighborhood of
\[\hat\varsigma(x,x_0)\cup \hat\varsigma(x',x'_1) \subset \hat\varsigma(x,z) \cup \hat\varsigma(x',z).\]
Thus each side of $\Delta^{\hat\varsigma}(x,x',z) = \Delta^{\hat\varsigma}(x,y,z)$ is contained in the $(2\delta'+2)$--neighborhood of the union of the other two, which proves the Lemma.
\end{proof}

\subsection{Proving that general triangles are thin}
We are now ready to prove Theorem~\ref{T:preferred slim triangles}, which again we do by decomposing a general triangle into simpler ones, this time of the previous type where one side of the reference triangle in the base fiber is a single saddle connection.

\begin{proof} [Proof of Theorem~\ref{T:preferred slim triangles}]  Let $\delta''>0$ be as provided by Lemma~\ref{L:length one side slim} and set $\delta = 3\delta''$.  Let $x,y,z \in \Sigma$ be any three points, and we must prove that $\Delta^{\hat \varsigma}(x,y,z)$ is $\delta$--slim.  By Lemma~\ref{L:excising degenerate parts}, we may assume that $\Delta(x,y,z)$ is nondegenerate.  We show that $\hat \varsigma(x,y)$ is in $N_\delta(\hat \varsigma(x,z) \cup \hat \varsigma(y,z))$.  If any side of $\Delta(x,y,z)$ is a saddle connection, this follows from Lemma~\ref{L:length one side slim}, so we suppose that all sides have a least two saddle connections.

We appeal to Lemma \ref{L:triangle decomposition} to decompose $\Delta^{\hat \varsigma}(x,y,z)$ into two or three triangles, depending on the configuration of geodesics from $f(x)$ to points in $[f(y),f(z)]$.  To describe the decomposition, recall that for each cone point $w$ on the side $[f(y),f(z)]$ of $\Delta(x,y,z) = \Delta(f(x),f(y),f(z))$, other than $f(y)$ and $f(z)$, the geodesic segment from $f(x)$ to $w$ is the concatenation of two geodesic segments $\nu\mu$, where:
\begin{enumerate}
\item $\nu$ is a geodesic subsegment of either $[f(x),f(y)]$ or $[f(x),f(z)]$, possibly a single point, and
\item $\mu$ is a single saddle connection with interior contained in the interior of $\Delta(x,y,z)$.
\end{enumerate}
See Figure~\ref{F:opposite sides} above.

If for some cone point $w$ in the interior of $[f(y),f(z)]$, $\nu$ degenerates to the point $f(x)$ (as in the left-most picture in Figure~\ref{F:opposite sides}), we subdivide $\Delta(x,y,z)$ into two triangles along the segment $[f(x),w]$. We use this to subdivide $\Delta^{\hat \varsigma}(x,y,z)$ into two triangles by choosing some point $\tilde{w}$ along the horizontal piece of the preferred path $\varsigma(y,z)$ with $f(\tilde{w}) = w$, and subdividing into $\Delta^{\hat \varsigma}(x,y,\tilde{w})$ and $\Delta^{\hat \varsigma}(x,z,\tilde{w})$. Since $\tilde{w}$ lies on the preferred path $\varsigma(y,z)$, we note that $\varsigma(y,z) = \varsigma(y,\tilde{w})\cup\varsigma(\tilde{w},z)$. Noting also that the two triangles $\Delta(x,y,\tilde{w}) = \Delta(f(x),f(y),w)$ and $\Delta(f(x),f(z),w)$ in the subdivision of $\Delta(x,y,z)$ are both nondegenerate with one side consisting of a single saddle connection, Lemma~\ref{L:length one side slim} implies that $\Delta^{\hat \varsigma}(x,y,\tilde{w})$ and $\Delta^{\hat \varsigma}(x,z,\tilde{w})$ are $\delta''$--slim. Thus two applications of Lemma~\ref{L:length one side slim} give the desired containment:
\begin{align*}
\hat\varsigma(x,y) &\subset N_{\delta''}\big(\hat\varsigma(y,\tilde{w})\cup \hat\varsigma(\tilde{w},x)\big)\\
&\subset N_{\delta''}\big(\hat\varsigma(y,\tilde{w})\big)\cup N_{2\delta''}\big(\hat\varsigma(\tilde{w},z)\cup \hat\varsigma(z,x)\big)\\
&\subset N_{2\delta''}\big(\hat\varsigma(y,z)\cup\hat\varsigma(z,x)\big).
\end{align*}

It remains to consider the case that for every cone point $w$ in the interior of $[f(y),f(z)]$, the geodesic does {\em not} consist of a single saddle connection as in the left-most picture of Figure~\ref{F:opposite sides}. Let $f(y) = w_0,w_1,\dotsc, w_k = f(z)$ be the sequence of cone points along the $[f(y),f(z)]$. By hypothesis, for each $i\in \{0,\dotsc, k\}$, the geodesic $[f(x),w_i]$ has nondegenerate intersection $\nu$ with either $[f(x),f(y)]$ or $[f(x),f(z)]$. We may thus partition the index set
\[\{0,\dotsc,k\} = Y\sqcup Z\]
according to whether this initial segment $\nu$ of $[f(x),w_i]$ lies in $[f(x), f(y)]$ (for $i\in Y$) or in $[f(x),f(z)]$ (for $i\in Z$). Notice $0\in Y$ and $k\in Z$ by construction. Furthermore, if $i\in Y$ then $j\in Y$ for all $j\le i$; indeed, this follows from the fact that the geodesic $[f(x),w_j]$ must lie in the convex subset $\Delta(f(x),f(y),w_i)$ of $\Delta(f(x),f(y),f(z))$. Therefore, there exists an index $0 \le i < k$ such that $Y = \{0,\dotsc, i\}$ and $Z = \{i+1,\dotsc, k\}$.

There are three cases to consider: Firstly, if $0 < i < k-1$, we choose points $\tilde{w}_i,\tilde{w}_{i+1}\in \varsigma(y,z)$ along the corresponding horizontal pieces of the preferred path so that $f(\tilde{w}_i) = w_i$ and $f(\tilde{w}_{i+1}) = w_{i+1}$. The preferred paths $\varsigma(x,y)$ and $\varsigma(x,\tilde{w}_i)$ then share a degenerate initial segment, and we let $\tilde{v_i}$ be the endpoint of this common initial segment; that is, $\varsigma(x,y)\cap \varsigma(x,\tilde{w}_i) = \varsigma(x,\tilde{v}_i)$. Similarly choose $\tilde{v}_{i+1}$ so that $\varsigma(x,z)\cap \varsigma(x,\tilde{w}_{i+1}) = \varsigma(x,\tilde{v}_{i+1})$. This decomposes $\Delta(x,y,z)$ into three nondegenerate triangles $\Delta(\tilde{v}_i,y,\tilde{w}_i)$, $\Delta(x,\tilde{w}_i,\tilde{w}_{i+1})$, and $\Delta(\tilde{v}_{i+1},\tilde{w}_{i+1},z)$, each of which having one side a single saddle connection. Therefore we may apply Lemma~\ref{L:length one side slim} three times to conclude the containment
\begin{align*}
\hat\varsigma(y,x) &= \hat\varsigma(y,,\tilde{v}_i)\cup \hat\varsigma(\tilde{v}_i,x)\\
&\subset N_{\delta''}\big(\hat\varsigma(y,\tilde{w}_i)\cup \hat\varsigma(\tilde{w}_i,\tilde{v}_i)\big)\cup \hat\varsigma(\tilde{v}_i,x)\\
&\subset N_{\delta''}\big(\hat\varsigma(y,\tilde{w}_i)\cup \hat\varsigma(\tilde{w}_i,x)\big)\\
&\subset N_{2\delta''}\big(\hat\varsigma(y,\tilde{w}_i)\cup \hat\varsigma(\tilde{w}_i,\tilde{w}_{i+1})\cup\hat\varsigma(\tilde{w}_{i+1},x)\big)\\
&= N_{2\delta''}\big(\hat\varsigma(y,\tilde{w}_{i+1})\cup \hat\varsigma(\tilde{w}_{i+1},\tilde{v}_{i+1})\cup\hat\varsigma(\tilde{v}_{i+1},x)\big)\\
&\subset N_{3\delta''}\big(\hat\varsigma(y,\tilde{w}_{i+1})\cup \hat\varsigma(\tilde{w}_{i+1},z)\cup \hat\varsigma(z,\tilde{v}_{i+1})\cup\hat\varsigma(\tilde{v}_{i+1},x)\big)\\
&= N_{3\delta''}\big(\hat\varsigma(y,z)\cup \hat\varsigma(z,x)\big).
\end{align*}

Secondly, if $i = 0$, we similarly choose a point $\tilde{w}_1\in \varsigma(y,z)$ along a horizontal piece such that $f(\tilde{w}_1) = w_1$. Since $1\in Z$ by hypothesis, the paths $\varsigma(x,z)$ and $\varsigma(x,\tilde{w}_1)$ share an initial segment and we again choose $\tilde{v}_1\in \varsigma(x,z)$ so that $\varsigma(x,z)\cap \varsigma(x,\tilde{w}_1) = \varsigma(x,\tilde{v}_1)$. This decomposes $\Delta(x,y,z)$ into two nondegenerate triangles $\Delta(x,y,\tilde{w}_1)$ and $\Delta(\tilde{v}_1,\tilde{w}_1,z)$ which each have a side consisting of a single saddle connection. Thus we may apply Lemma~\ref{L:length one side slim} twice, as above, to obtain
\begin{align*}
\hat\varsigma(y,z) &\subset N_{\delta''}\big(\hat\varsigma(y,\tilde{w}_1)\cup\hat\varsigma(\tilde{w}_1,x)\big)\\
&= N_{\delta''}\big(\hat\varsigma(y,\tilde{w}_1)\cup\hat\varsigma(\tilde{w}_1,\tilde{v_1})\cup \hat\varsigma(\tilde{v}_1,x)\big)\\
&\subset N_{2\delta''}\big(\hat\varsigma(y,\tilde{w}_1)\cup\hat\varsigma(\tilde{w}_1,z)\cup\hat\varsigma(z,\tilde{v_1})\cup \hat\varsigma(\tilde{v}_1,x)\big)\\
&= N_{2\delta'}\big(\hat\varsigma(y,z)\cup\hat\varsigma(z,x)\big),
\end{align*}
showing that $\Delta^{\hat\varsigma}(x,y,z)$ is $2\delta''$--slim. The remaining case $i = k-1$ is handled symmetrically by choosing $\tilde{w}_{k-1}\in \varsigma(y,z)$ with $f(\tilde{w}_{k-1}) = w_{k-1}$, and choosing $\tilde{v}_{k-1}$ so that $\varsigma(x,y)\cap \varsigma(x,\tilde{w}_{k-1}) = \varsigma(x,\tilde{v_{k-1}})$. This decomposes $\Delta(x,y,z)$ into two triangles $\Delta(y,\tilde{v}_{k-1},\tilde{w}_{k-1})$ and $\Delta(x,\tilde{w}_{k-1},z)$ to which we may apply Lemma~\ref{L:length one side slim} and conclude $\hat\varsigma(x,y)\subset N_{2\delta''}\big(\hat\varsigma(y,z)\cup\hat\varsigma(z,x)\big)$ as above. This covers all the cases and completes the proof of Theorem~\ref{T:preferred slim triangles}.
\end{proof}

\subsection{Loxodromic elements}
Recall that an isometry $\phi$ of a hyperbolic space $Y$ is \emph{loxodromic} if its \emph{translation length}
\[t(\phi) = \lim_{n\to \infty} \frac{d(y, \phi^n(y))}{n}\quad\text{for any}\quad y\in Y\]
is positive. Now that we know $\hat E$ is hyperbolic, we can characterize which elements of $\Gamma$ act loxodromically on $\hat E$. The characterization is quite simple:

\begin{proposition}\label{prop:loxodromics}
An element $\gamma \in \Gamma$ acts loxodromically on $\hat E$ if and only if $\gamma$ does not stabilize any vertex $v\in \vtx$.
\end{proposition}

\begin{proof}
Obviously $\gamma$ cannot be loxodromic if it fixes a point. Conversely, suppose $\gamma$ does not fix any point of $\vtx$ and let $g\in G$ be the image of $\gamma$ in the quotient. If $g$ is a pseudo-Anosov element of $G\le \Mod(S)$, then $g$ acts with positive translation length on $\hat D$ and therefore $\gamma$ acts with positive translation length on $\hat E$ (since $\hat \pi\colon \hat E\to \hat D$ is Lipschitz).
Hence, after passing to a power if necessary, we may assume that $g$ is parabolic or trivial in $G$. Either way, we may choose some $\alpha \in \CP$ that is fixed by $g$.

It follows that $\gamma$ preserves the subsets $\partial \CB_\alpha\subset \bar E$ and $T_\alpha \subset \hat E$. Since $\gamma$ does not stabilize any vertex of $T_\alpha$, it restricts to a loxodromic isometry of the tree $T_\alpha$. Let
\[\omega = \dotsb\omega_{-2}\omega_{-1}\omega_0\omega_1\omega_2\dotsb\]
be the bi-infinite axis of $\gamma$ in $T_\alpha$, viewed as an edge path in which each $\omega_i$ denotes an edge of $T_\alpha$ between, say, vertices $v_{i-1},v_i\in T_\alpha^{(0)}\subset \vtx$.

For any $X\in \partial B_\alpha$ and $i\in \mathbb Z$, let $\Omega^i_X \subset E_X$ be the closure of the preimage of the interior of the edge $\omega_i$ under $P\vert_{E_X}$. Thus $\Omega^i_X$ is a closed strip in direction $\alpha$ that is foliated by lines in direction $\alpha$ and separates $E_X$ into two pieces. Also let
\[\Omega_X = \bigcup_i \Omega^i_X\]
be the union of these strips.  Each component of $\Omega_X$ is a totally convex subset of $E_X$ whose boundary components are each contained in a spine $\theta^{v_i}_X$ for some $i\in \mathbb Z$.

Let $r\colon \partial \CB_\alpha\to \omega$ be the composition of $P\vert_{\partial \CB_\alpha}\colon \partial \CB_\alpha\to T_\alpha$ with the closest-point-projection $T_\alpha\to \omega$. Since $P$ is $\Gamma$--equivariant and $\gamma$ is an $\omega$--preserving isometry of $T_\alpha$, this map $r$ is equivariant with respect to the action of the cyclic group $\langle \gamma \rangle$; that is, $r(\gamma x) = \gamma r(x)$.
We note that $r\vert_{E_X} = r \circ f_{X,Y}$ for any $X,Y\in \partial B_\alpha$, which follows from the fact that in this case $P\vert_{E_Y} = P\circ f_{X,Y}$.
Next define a map
\[r_\omega\colon \vtx \to \mathbb P(\omega)\quad\text{given by}\quad
r_\omega(v) = r(\theta^v_X)\subset \omega\quad\text{for any}\quad X\in \partial B_\alpha,\]
which is well-defined since $\theta^v_X = f_{X,Y}(\theta^v_Y)$. Furthermore, since $\gamma \theta^v_X = \theta^{\gamma v}_{g X}$, we have
\[\gamma r_\omega(v) = \gamma r(\theta^v_X) = r(\gamma \theta^v_X) = r(\theta^{\gamma v}_{g X}) = r_\omega(\gamma v).\]
That is, $r_\omega$ is equivariant with respect to the actions of $\gamma$ on $\vtx$ and $\omega$.

The heart of the proposition is captured in the following claim:

\begin{claim}
\label{claim:loxo-spine-diam-bound}
There exists  $M> 0$ such that $\diam_{T_\alpha}(r_\omega(v)) \le M$ for all $v\in \vtx$.
\end{claim}

Assuming the claim, let us deduce the proposition. For $u,v\in \vtx$, define
\[d_\omega(u,v) = \diam_{T_\alpha}(r_\omega(u)\cup r_\omega(v)).\]
If $u,v$ are connected by a horizontal jump in $\hat E$, then the spines $\theta^u_X$ and $\theta^v_X$ intersect for any $X\in D$. Thus by the claim $d_\omega(u,v)\le 2M$. By Lemma~\ref{L: combinatorial path}, there is some $C$ such that any two vertices $u,v\in \vtx$ may be connected by a combinatorial path consisting of at most $C\hat d(u,v)$ jumps. Therefore
\[d_\omega(u,v) \le 2MC\hat d(u,v)\quad\text{for all}\quad u,v\in \vtx.\]
Finally, if $t(\gamma) > 0$ denotes the translation length of $\gamma$ acting on $T_\alpha$, then by definition for all $v\in\vtx$ and $n\ge 0$ the equivariance $r_\omega(\gamma^n v) = \gamma^n r_\omega(v)$ implies that
\[ 2MC\hat d(v, \gamma^nv )\ge  d_\omega(v,\gamma^n v) = \diam_{T_\alpha}(r_\omega(v) \cup \gamma^n r_\omega(v)) \ge n t(\gamma).\]
Thus $\gamma$ indeed acts loxodromically on $\hat E$ as claimed.

It remains to prove Claim~\ref{claim:loxo-spine-diam-bound}. Let us first describe the restriction of $r$ to $E_X$ for $X\in \partial B_\alpha$. Since $P$ maps $\Omega_X$ to $\omega$, we see that $r$ agrees with $P$ on $\Omega_X$. For each component $U$ of $E_X\setminus \Omega_X$, there exist some $i\in \mathbb Z$ such that $r(\overline{U}) = \{v_i\}$ with $\partial \overline{U}\subset \theta^{v_i}_X$. From this it follows that if $r_\omega(v)$ is not a single point, then $\theta^v_X\cap \Omega_X\ne \emptyset$ and moreover that  $r(\theta^v_X\cap \overline{U}) = r(\theta^v_X\cap \partial \overline{U})$ for every such component $U$.
Since $\partial \overline{U}\subset \Omega_X$, we conclude $r_\omega(v) = r(\theta^v_X\cap \Omega_X)$ in this case. Hence, in Claim~\ref{claim:loxo-spine-diam-bound} it suffices to bound $\diam(r(\theta^v_X\cap \Omega_X))$.

We also note that when $r_\omega(v)$ is not a singleton, $\theta^v_X\cap \Omega_X$ is contained in a convex component of $\Omega_X$. Indeed, the convex set $(P\vert_{E_X})^{-1}(\omega)$ is the union of $\Omega_X$ with $\cup_{i\in \mathbb Z} \theta^{v_i}_X$. Thus if $\theta^v_X$ were to intersect distinct components of $\Omega_X$, then $\theta^v_X$ must intersect some $\theta^{v_i}_X$ in a saddle connection. But that implies $\alpha(v) = \alpha(v_i) = \alpha$ and therefore that $P(\theta^v_X)$ is a single point, contrary to our assumption on $v$.

We now complete the proposition by proving Claim~\ref{claim:loxo-spine-diam-bound} in two cases: \qedhere

\begin{proof}[Proof of Claim~\ref{claim:loxo-spine-diam-bound} when $g =1$.]
Fix any $X\in \partial B_\alpha$. In this case $\gamma \Omega_X = \Omega_{gX} = \Omega_X$, so that $\gamma$ gives an isometry of $E_X$ that preserves the convex set $\Omega_X$.

The strips $\Omega^i_X$ limit, as $i\to \pm\infty$, on two points in the $\mathrm{CAT}(0)$ boundary of $E_X$. Let $\ell$ denote the unique geodesic joining these boundary points, which is a $\gamma$--invariant, bi-infinite geodesic that crosses each strip $\Omega^i_X$.
Note that $\ell$ cannot lie in any spine $\theta^v_X$, since then $\gamma$ would fix  $v\in \vtx$. Therefore the saddle connections comprising $\ell$ do not  all have the same direction.
Write
\[\ell =  \ldots \sdl_{-2} \sdl_{-1} \sdl_0 \sdl_1 \sdl_2 \ldots \]
where each $\sdl_i$ is a maximal concatenation of saddle connections in a single direction. There exists $\mu > \pi$ so that $\sdl_i$ and $\sdl_{i+1}$ form an angle of least $\mu$ on both sides.

Consider any vertex $v\in \vtx$. By the above discussion, it suffices to take any points $y,z\in \theta^v_X\cap \Omega_X$ with $r(y)\ne r(z)$.
Let $U$ be the convex component of $\Omega_X$ containing $\theta^v_X\cap \Omega_X$.
Choose $i\in \mathbb Z$ so that $y\in \Omega^i_X$, and let $y'$ be the unique point of $\ell\cap \Omega^i_X$ with $r(y') = y$.  That is, $y'$ is the intersection of $\ell$ with the leaf $\mathcal{F}_y$ through $y$ of the foliation of $\Omega^i_X$ in direction $\alpha$ (note that $\ell\cap \Omega^i_X$ is a segment in a direction distinct from $\alpha$). Choose $z'\in \ell$ and $\mathcal{F}_z$ similarly. Finally, let $\eta\subset \theta^v_X$ be the geodesic from $y$ to $z$ and $\ell_0\subset \ell$ the geodesic from $y'$ to $z'$.

If $\ell_0$ and $\eta$ are disjoint, then the geodesics $\mathcal{F}_y,\eta,\mathcal{F}_z,\ell_0$  form a geodesic quadrilateral in $U$ with no cone points in its interior (since there are no cone points in the interior of $\Omega_X$). Otherwise $\ell_0$ and $\eta$ intersect and we get two geodesic triangles connected by the (possibly degenerate) segment $\ell_0\cap \eta$. By Gauss--Bonnet, doubling this picture produces an object $W$ that is either a sphere with total curvature $4\pi$ (in the quadrilateral case) or two spheres (when $\ell_0\cap \eta\ne\emptyset$) with total curvature $8\pi$. The only points of positive curvature come from the four corners $y,y',z,z'$ and the two endpoints of $\ell_0\cap \eta$ (when it exists). Further, each of these points contributes positive curvature at most $2\pi$. On the other hand, each cone point juncture $\sdl_{j}\sdl_{j+1}$ in the interior of $\ell_0\setminus \eta$ contributes negative curvature at most $2(\pi -\mu) < 0$ to $W$. Note also that $\ell_0\cap \eta$ is necessarily contained in a single segment $\sigma_j$ by construction, since all of $\eta$ lies in the single direction $\alpha(v)$. Therefore, if $m$ denotes the number of segments $\sdl_j$ that have nondegenerate intersection with $\ell_0$ we conclude that
\[
4\pi \le 2\pi \chi(W) \le 6(2\pi)  + 2(\pi -\mu)(m-3) \implies m\le \frac{4\pi}{\mu-\pi} + 3.\]
Since $\ell$ is $\gamma$--invariant, there is some maximum number $N$ of strips crossed by any segment $\sdl_i$. Thus $\ell_0$ crosses at most $3N + 4N\pi/(\mu - \pi)$ strips, and we are done.
\end{proof}

\begin{proof}[Proof of Claim~\ref{claim:loxo-spine-diam-bound} when $g\ne 1$]
In this case $g$ is a parabolic element of the Veech group $G = G_q$ fixing the direction $\alpha \in \mathbb{P}^1(q)$. Moreover, the iterates $g^m$ converge uniformly on compact subsets of $\mathbb P^1(q)\setminus\{\alpha\}$ to the constant map $\mathbb P^1(q)\setminus\{\alpha\}\mapsto \{ \alpha\}$.

Note that any geodesic connecting the two boundary components of a strip $\Omega^i_X$ is a straight segment in a single direction with no cone points in its interior. Thus for $i\in \mathbb Z$ we are justified in defining $A^i_X\subset\mathbb P^1(q)$ to be the set of directions of all segments of the form $\ell\cap \Omega^i_X$, where $\ell$ is any geodesic segment from $\Omega^{i-1}_X$ to $\Omega^{i+1}_X$. Note that such a segment $\ell\cap \Omega^i_X$ has it endpoints in the closest-point-projections of $\Omega^{i\pm 1}_X$ to $\Omega^i_X$. Since such a closest-point-projection is equal to a saddle connection or cone point in $\partial \Omega^i_X$, we see that $A^i_X$ is a compact subset of $\mathbb P^1(q)\setminus \{\alpha\}$.

Let $k\in \mathbb N$ be such that $\gamma \omega_i = \omega_{i+k}$ for all $i\in \mathbb Z$. Thus for any $X\in \partial B_\alpha$ we have $\gamma \Omega^{i}_X= \Omega^{i+k}_{gX}$.
By definition, it follows that $g A^i_X = A^{i+k}_{gX}$. Since the affine maps $f_{Y,Z}\colon E_Z\to E_Y$ fix the directions of lines (viewed in  $\mathbb P^1(q)$), we conclude that
\[A^{i+k}_X = f_{X,gX}(A^{i+k}_{gX}) = g A^{i}_X.\]
By this and the uniform convergence of the maps $\{g^m\}$ on $\mathbb P^1(q)\setminus\{\alpha\}$, we may choose $m\ge 1$ such that $A^{i+km}_X  = g^m A^i_X$ and $A^i_X$ are disjoint for all $i\in \mathbb Z$.

To prove the claim, consider any $v\in \vtx$. Note that all saddle connections of $\theta^v_X$ lie in one fixed direction $\alpha(v)$. Suppose $\theta^v_X$ intersects strips $\Omega^{i-1}_X$ and $\Omega^{i+km+1}_X$. It follows that $\theta^v_X$ intersects every strip $\Omega^j_X$ with $i\le j\le i+km$, since these each separate $\Omega^{i-1}_X$ from $\Omega^{i+km+1}_X$. Therefore the direction $\alpha(v)$ lies in both $A^i_X$ and $A^{i+km}_X$, contradicting our choice of $m$. This proves that $\theta^v_X$ can intersect at most $km+2$ strips and  bounds $\diam_{T_\alpha}(r_\omega(v))$.
\end{proof}
\end{proof}

\section{Projections and vertex spaces} \label{S:projections}

An HHS structure on a metric space consists of certain additional data, most importantly a collection of hyperbolic spaces together with projection maps to each space. For the HHS structure that we will build on (Cayley graphs of) $\Gamma$, the hyperbolic spaces will (up to quasi-isometry) be $\hat E$ and the spaces $\mathcal K^v$ and $\Xi^v$ introduced in this section, where $v$ varies over all vertices of the trees $T_\alpha$.
Morally, the projections will be given by the maps $\Lambda^v$ and $\xi^v$ that we study below. However, to prove hierarchical hyperbolicity we will use a criterion from \cite{comb_HHS} which does not require actually defining projections, but nevertheless provides them. Still, the maps $\Lambda^v$ and $\xi^v$ will play a crucial role in proving this criterion applies.

We will establish properties of $\Lambda^v$ and $\xi^v$ that are reminiscent of subsurface projections or of closest-point projections to peripheral sets in relatively hyperbolic spaces/groups; these are summarized in Proposition \ref{prop:R-projections}.
Essentially, these same properties would be needed if we wanted to construct an HHS structure on $\Gamma$ directly without using \cite{comb_HHS}.

From a technical point of view, we would like to draw attention to Lemma \ref{lem:window}, which is the crucial lemma that ensures the projections behave as desired and that various subspaces have bounded projections. Roughly, the lemma says that closest-point projections to a spine does not vary much under affine deformations.

\subsection{Notation reminders}
Before we begin, it is useful to recall the definition of the vertex set $\vtx$ and associated notation $\alpha \colon \vtx \to \CP$, $d_{\tree}$, etc. from \S\ref{S:tree-vertices}, as well as the spines $\theta^v_X$,  thickened spines $\bTheta^v_X$, and the various associated bundles
\[ \theta^v \subset \bTheta^v \subset  \partial \mathcal B_v = \partial \mathcal B_{\alpha(v)}\]
over $\partial B_v = \partial B_{\alpha(v)}$ from \S\ref{S:spines}.  We will write $d_{\bTheta^v}$ and $d_{\partial {\mathcal B}_\alpha}$ for the induced path metrics on $\bTheta^v$ and $\partial {\mathcal B}_\alpha$ from $\bar d$.  Using the map $f_\alpha \colon \bar E \to \partial {\mathcal B}_\alpha$, it is straightforward to see that $d_{\partial {\mathcal B}_\alpha}$ is uniformly coarsely equivalent to the subspace metric from $\bar d$: in fact, $\bar d \leq d_{\partial {\mathcal B}_\alpha} \leq e^{\bar d}\bar d$.  The same is true for $d_{\bTheta^v}$, which follows from the fact that the inclusion of $\bTheta^v$ into $\partial {\mathcal B}_\alpha$ is a quasi-isometric embedding with respect to the path metrics (see below).

\iffalse
\begin{definition}
For $v,w \in \vtx$, we write $v \parallel w$ if $\alpha(v) = \alpha(w)$. We then define  $d_{\tree}(v,w) \in \mathbb Z_{\geq 0} \cup \{\infty\}$ to be the combinatorial distance in the simplicial tree $T_{\alpha(v)}= T_{\alpha(w)}$ when $v \parallel w$  (as opposed to the distance from the preferred $\mathbb R$--tree metric) and to
equal $\infty$ when $v\not\parallel w$.
\end{definition}

For any $\alpha \in \CP$ and $X \in D$, we consider the union of the set of all saddle connections in direction $\alpha$ in $E_X$.  The components of this space are precisely the preimages of vertices of $v \in T_\alpha$ under the map $P \circ f_{X_\alpha,X} \colon E_X \to T_\alpha$.  For a vertex $v \in T_\alpha^{(0)}\subset \vtx$ and $X \in D$, we write $\theta^v_X = (P \circ f_{X_\alpha,X})^{-1}(v) \subset E_X$ for this component, which we call the {\em $v$--spine in $E_X$}.  The closure of each component of
\[ E_X -  \bigcup_{v \in T_\alpha^{(0)}} \theta^v_X \]
is an infinite strip covering a Euclidean cylinder in direction $\alpha$.  For each $v$, let $\bTheta^v_X$ be the union of $\theta^v_X$ together with all these infinite strips that meet $\theta^v_X$, which we call the {\em thickened $v$--spine in $E_X$}.  Note that $f_{X,Y}$ maps $\theta^v_Y$ and $\bTheta^v_Y$, respectively, to $\theta^v_X$ and $\bTheta^v_X$, respectively.

The union of the $v$--spines and thickened $v$--spines over $\partial B_v$ are denoted
\[ \theta^v = \bigcup_{X \in \partial B_v} \theta^v_X \quad \quad \bTheta^v = \bigcup_{X \in \partial B_v} \bTheta^v_X. \]
These are bundles over $\partial B_v$ which we call the {\em $v$--spine bundle} and the {\em thickened $v$--spine bundle}.

\sd{I think we decided to cut this remark? Maybe something related to this will be said in the intro?}
\begin{remark} The space $\hat E$ is quasi-isometric to the space obtained from $\bar E$ by coning off each space $\theta^v$ (or coning off each space $\bTheta^v$).  Although this coning-off of $\bar E$ may be simpler, the trees $T_\alpha$ are fundamental objects associated to the foliations, and it seems more natural to work with $\hat E$.
\end{remark}
\fi

Associated to each $v \in \vtx$ we will be considering two types of projections.  These projections are related by a single projection $\Pi^v \colon \bar{\Sigma} \to \bTheta^v$.  It is convenient to analyze $\Pi^v$  via an auxiliary map which serves as a kind of fiberwise closest point projection that survives affine deformations, and which we call the {\em window map}.  We begin by describing the two types of projections restricted to $\bTheta^v$, and explain some of their basic features.  Next we define the window map and prove what is needed from it. Finally, we define $\Pi^v$ and prove the key properties of the associated projections.

\subsection{Kapovich-Leeb distances} \label{S:Kapovich-Leeb distances}

For each $v\in \vtx$, we will use the work of Kapovich and Leeb  \cite{KapovichLeeb} (specifically, Theorem~\ref{T:KL theorem} below) to define a \emph{level map}
\[\lambda^v \colon \bTheta^v \rightarrow \mathcal K^v,\]
where $\mathcal K^v$ is a discrete set isometric to $\mathbb Z$ with the standard metric $|x - y|$.  Roughly, for each $v$, from Theorem~\ref{T:KL theorem}, one obtains a coarse product structure on $\bTheta^v$ as the product $\bTheta^v_X \times \mathbb R$, and $\lambda^v$ is the integer part of the projection to the $\mathbb R$--factor.  The relevant properties of $\lambda^v$ are given by the next proposition. We note that the proposition and Lemma \ref{lem:retract_on_the_middle} can be used as black-boxes (in particular, the definitions of $\lambda^v$ and $\mathcal K^v$ are never used after we prove those results).

\begin{proposition}\label{prop:KL}
There exist $K_1>0$ such that for each $v \in T_\alpha^{(0)} \subset \vtx$, there exists a map $\lambda^v\colon \bTheta^v \rightarrow \mathcal K^v$ satisfying the following properties:
\begin{enumerate}
\item \label{item:lambda_lipschitz} $\lambda^v$ is $K_1$--coarsely Lipschitz with respect to the path metric on $\bTheta^v$;
\item \label{item:lambda_flowlines_bounded} For any $x \in \partial \bTheta^v$, if $\ell_{x,\alpha} = D_x \cap \partial \mathcal B_\alpha$ then $\lambda^v(\ell_{x,\alpha})$ is a singe point.
\item\label{item:KL_qi} For any $v,w\in \vtx$ with $d_{\mathrm{tree}}(v,w)=1$, $\lambda^v\times\lambda^w\colon \bTheta^v\cap \bTheta^w \to \mathcal K^v\times \mathcal K^w$ is a surjective $(K_1,K_1)$--quasi-isometry with respect to the induced path metric on the domain.
\end{enumerate}
\end{proposition}

To explain the proof it is useful to review some background on graph manifolds, which we do now.\\

\noindent {\bf Coarse geometry of graph manifolds}.
Recall that a graph manifold is a $3$--manifold that contains a finite union of tori, so that cutting along the tori produces a disjoint union of Seifert fibered $3$--manifolds, called the {\em Seifert pieces}.  Seifert fibered $3$--manifolds are compact $3$--manifolds foliated by {\em circle leaves}.  For Theorem~\ref{T:KL theorem} below, we exclude Seifert manifolds and Sol-manifolds in the definition of graph manifold.
A {\em flip manifold} is a graph manifold so that every Seifert piece is a product of a circle and a compact surface, with the property that if two Seifert pieces share a common torus boundary component, the circle leaves of one match with the boundaries of surface fibers of the other along that boundary component, and vice versa.  That is, all identifications of boundary tori interchange the foliations by circle leaves with the foliations by boundaries of surface fibers.
Any flip manifold admits a nonpositively curved Riemannian metric for which each product Seifert piece is given a product metric with totally geodesic boundary, and each of which is locally isometrically embedded with respect to those metrics.  We call such a metric on a flip manifold a {\em standard nonpositively curved metric}.

The universal cover of a graph manifold decomposes into a union of universal covers of the Seifert pieces glued together along $2$--planes (covering the tori).  The decomposition is dual to a tree, and the universal covers of the Seifert pieces are {\em the vertex spaces}.  For any Seifert fibered space, its universal cover is foliated by lines, the lift of the foliation by circles, and we refer to the leaves simply as {\em lines} in the universal cover.  The next result describes the key coarse geometry feature of the universal cover of a graph manifold we will need; see \cite{JacoShalen} and \cite{KapovichLeeb}.

\begin{theorem} [Kapovich--Leeb] \label{T:KL theorem} The universal covering of any graph manifold, equipped with the lift of any geodesic metric, admits a quasi-isometric homeomorphism to the universal covering of a flip manifold with the lift of a standard nonpositively curved metric.  Moreover, this quasi-isometry preserves the decomposition into vertex spaces, and sends lines to lines on each vertex space.
\end{theorem}

\noindent {\bf Horocycles and bundles}.
Next we describe the specific graph manifolds that are relevant for our purposes.

Let $G_\alpha < G$ denote the stabilizer of $B_\alpha$, for each $\alpha \in \mathcal P$.  This has a finite index cyclic subgroup generated by a multitwist, $\langle \tau_\alpha \rangle <_{f.i.} G_\alpha$; see \S\ref{S:Veech groups}.  The preimage of $G_\alpha$ in $\Gamma$ is the $\pi_1S$--extension group $\Gamma_\alpha$ of $G_\alpha$.   The action of $\Gamma_\alpha$ on $\partial \mathcal B_\alpha$ is cocompact, and the quotient is the graph manifold  mentioned in the introduction.  Since we are interested in the geometry of $\partial \mathcal B_\alpha$ (and not so much the specific compact quotient), we pass to a finite index subgroup and assume $G_\alpha = \langle \tau_\alpha \rangle$ for simplicity.  The manifold $\partial \mathcal B_\alpha/\Gamma_\alpha$ is the mapping torus of $\tau_\alpha$, and we describe its graph manifold structure in more detail.

Consider the surface $S$ with the flat metric $q_{X_\alpha}$, so that $(S,X_\alpha,q_{X_\alpha}) = E_{X_\alpha}/\pi_1S$.  The multitwist $\tau_\alpha$ is an affine map that preserves the cylinders in direction $\alpha$, acting as a power of a Dehn twist in each cylinder and as the identity on their boundaries.  The union of the boundaries of the cylinders are spines (deformation retracts) for the subsurfaces that are the complements of the twisting curves (core curves of the cylinders).  Consequently, $\tau_\alpha$ is the identity on these spines.  The homeomorphism $\tau_\alpha$ induces a homeomorphism on the subsurface obtained by cutting open $S$ along a core curve of each cylinder.  Each such induced homeomorphism is the identity on the corresponding spine, and is thus isotopic to the identity relative to the spine.  The mapping tori of each subsurface is a product of the subsurface times a circle, and embeds in the the mapping torus $\partial \mathcal B_\alpha/\Gamma_\alpha$ of $\tau_\alpha$.  These sub-mapping tori are the Seifert pieces for the graph manifold structure on $\partial \mathcal B_\alpha/\Gamma_\alpha$.

The lifted graph manifold decomposition of $\partial \mathcal B_\alpha$ corresponds to $T_\alpha$.  That is, for each $v \in T_\alpha^{(0)}$, there is a vertex space contained in $\bTheta^v$ and containing $\theta^v$.  In fact, with respect to the covering group, $\bTheta^v$ is an invariant bounded neighborhood of the vertex space and $\theta^v$ is an equivariant deformation retraction of that space.
The suspension flow on the mapping torus $\partial \mathcal B_\alpha/\Gamma_\alpha$ restricted to each spine defines circle leaves of the corresponding Seifert piece; that is, flow lines through any point on the spine are precisely the circle leaves.  In the universal covering $\partial \mathcal B_\alpha$, the lifted flowline through a point $x \in \partial \mathcal B_\alpha$ is a lifted horocycle, $\ell_{x,\alpha} = D_x \cap \partial \mathcal B_\alpha$.  Thus, for any vertex $v$ and any $x \in \theta^v$, $\ell_{x,\alpha}$ is a line for the vertex space corresponding to $v$.

\begin{remark}
One caveat about the lines for the vertex spaces: flowlines through points {\em not} on a spine are {\em not} lines of any vertex space.  In fact, they are not even uniformly close to lines for any vertex space.
\end{remark}

\noindent {\bf The level map}.
We are now ready to define the level map and prove the main properties we will need about it.
\begin{proof}[Proof of Proposition~\ref{prop:KL}] First, let $F \colon \partial \mathcal B_\alpha \to \flip$ be a quasi-isometric homeomorphism, as given by Theorem~\ref{T:KL theorem}, from $\partial \mathcal B_\alpha$ (with its induced path metric) to the universal cover $\flip$ of a flip manifold.  The space $\bTheta^v$ is a uniformly bounded neighborhood of a vertex space of $\partial \mathcal B_\alpha$ corresponding to $v$, and so maps by $F$ to (a uniformly bounded neighborhood of) the corresponding vertex space of $\flip$, which is a product $Z_v \times \mathbb R$, where $Z_v$ is the universal cover of a compact surface with boundary. We can define $\lambda^v$ to be the composition of $F|_{\bTheta_v}$, the closest point projection to $Z_v \times \mathbb R$, and the integer part of the map to the $\mathbb R$-factor. In particular, $\lambda^v$ is a composition of coarsely Lipschitz maps, so it is coarsely Lipschitz,  proving item (\ref{item:lambda_lipschitz}).

For item (\ref{item:lambda_flowlines_bounded}), note that since $x \in \partial \bTheta^v$, there exists a vertex $w \in T_\alpha^{(0)}$ with $d_\tree(v,w) = 1$ so that $x \in \theta^w$.  Therefore, $\ell_{x,\alpha} = D_x \cap \partial \mathcal B_\alpha$ is a line for the vertex space for $w$, and consequently, $F(\ell_{x,\alpha})$ is a line for the vertex space $Z_w \times \mathbb R$ in $\flip$.  The projection to $Z_v \times \mathbb R$ is just the closest point projection to the flat of intersection $\Phi = (Z_w \times \mathbb R) \cap (Z_v \times \mathbb R)$, which is a boundary flat of $Z_w \times \mathbb R$.  Since $Z_w \times \mathbb R$ is endowed with a product metric, the image of $\ell_{x,\alpha}$ by the closest point projection to this flat is another line $\{x' \} \times \mathbb R$.  Since $\flip$ is a flip manifold, this is the boundary of a surface factor of $Z_v \times \mathbb R$, and hence projects to a point in $\mathcal K^v$, as required.

Regarding item (\ref{item:KL_qi}), note that $\bTheta^v\cap \bTheta^w$ is a bounded neighborhood of the flat covering the torus between the Seifert pieces corresponding to $v$ and $w$.   Therefore $F(\bTheta^v\cap \bTheta^w)$ is a bounded neighborhood of the flat $\Phi = (Z_v \times \mathbb R) \cap (Z_w \times \mathbb R)$.  The map $\lambda^v\times\lambda^w$ is coarsely the composition of the restriction of $F$ with the map to the factors of a product structure $\Phi \cong \mathbb R\times \mathbb R$, yielding item (\ref{item:KL_qi}).
\end{proof}

The following technical lemma will be needed in \S\ref{S:combinatorial HHS}, but we prove it here since we have now established the setup for its proof.
For any $s\in \mathcal K^v$ we write
\[ M(s)=(\lambda^v)^{-1}(s) \subset \bTheta^v.\]

\begin{lemma}\label{lem:retract_on_the_middle}
   There is a function $N = N_{\mathcal K} \colon [0,\infty)\to[0,\infty)$ with the following property. Suppose that $v,v_1,v_2\in\vtx$ are so that $d_\tree(v,v_i)=1$. Then for each $s\in\mathcal K^v$ and $t_i\in\mathcal K^{v_i}$ we have
   $$\bar d(M(s)\cap M(t_1),M(s)\cap M(t_2))\leq N(\bar d(M(t_1),M(t_2))).$$
\end{lemma}

 \begin{proof}
It suffices to prove the lemma with $\bar d$ replaced by the path metric $d_{\partial \mathcal B_\alpha}$ on $\partial \mathcal B_\alpha$, since they are uniformly coarsely equivalent.  Next, observe that the quasi-isometric homeomorphism $F \colon \partial \mathcal B_\alpha \to \flip$ to the flip manifold $\flip$ from the proof of Proposition~\ref{prop:KL} (given by Theorem~\ref{T:KL theorem}) maps $M(s)$ and $M(t_i)$ to   uniformly bounded neighborhoods of the surface factors $Z(s) = Z_v \times \{s\}$ and $Z(t_i) = Z_{v_i} \times \{t_i\}$ in the vertex spaces $Z_v \times \mathbb R$ and $Z_{v_i} \times \mathbb R$, respectively, for $i = 1,2$.  Moreover, $F$ maps the intersection $M(s) \cap M(t_i)$ to a uniformly bounded neighborhood of the point $Z(s) \cap Z(t_i)$, for $i=1,2$, so it suffices to prove the analogous statements for these intersections in $\flip$.  In fact, we will prove
\[ d_\flip (Z(s) \cap Z(t_1), Z(s) \cap Z(t_2)) = d_\flip (Z(t_1),Z(t_2)), \]
which immediately implies the lemma.

For this, we recall that $\flip$ is equipped with a CAT(0) metric so that each vertex space $Z_w \times \mathbb R$ isometrically embeds as a product metric.  The closest point projection $r \colon \flip \to Z_v \times \mathbb R$ is a $1$--Lipschitz retraction, and because $\flip$ is a flip manifold, $r(Z(t_i))$ has the form $\{p_i\}\times \mathbb R$ for some $p_i \in \partial Z_v$, for each $i$.  Indeed, for each $i$ we have $r(Z(t_i)) = Z(t_i) \cap (Z_v \times \mathbb R)$.
 It follows that $d_{\flip}(Z(t_1),Z(t_2)) = d_{\flip}(r(Z(t_1)),r(Z(t_2)))$, and since $Z(s) \subset Z_v \times \mathbb R$ is a surface factor, we have that
\[ d_{\flip}(Z(s) \cap Z(t_1), Z(s) \cap Z(t_2)) = d_\flip (r(Z(t_1)),r(Z(t_2)),\]
which completes the proof.
\end{proof}

\subsection{Coned-off surfaces}

For $v\in \vtx$, we define $\bXi^v$ to be the graph whose vertices are all $w\in\vtx$ so that $d_\tree(v,w)=1$, and with edges connecting the pairs $w,w'$ whenever $\bTheta^w\cap \bTheta^{w'}\neq \emptyset$.
As such vertices $w\in \vtx$ are in bijective correspondence with the boundary components of $\bTheta^v$, there is an ``inclusion'' map
\[i^v\colon \partial \bTheta^v \to \bXi^v\]
that sends any point $x \in \partial \bTheta^v$ to the vertex $w$ for which $x \in \bTheta^w$.
In light of the following lemma, we note that we could alternately define the edges of $\bXi^v$ in terms of subspaces lying within bounded distance of each other, and produce a space quasi-isometric to $\bXi^v$.

\begin{lemma}\label{lem:parameter_for_Xi}
There is a function $N = N_{\bXi} \colon[0,\infty)\to [0,\infty)$ so that whenever $v,w_1,w_2\in \vtx$ satisfy $\bar d(\bTheta^{w_1}\cap \bTheta^v,\bTheta^{w_2}\cap \bTheta^v)\leq r$,
the sets $\bTheta^{w_1}\cap \bTheta^v$ and $\bTheta^v\cap\bTheta^{w_2}$ may be connected via a concatenation of at most $N(r)$ paths, each of which is contained in a set of the form $\bTheta^v\cap \bTheta^{w}$.
\end{lemma}

\begin{proof}
If $\bar d(\bTheta^{w_1}\cap \bTheta^v,\bTheta^{w_2}\cap \bTheta^v)\leq r$, then there are cone points $p_i\in\bTheta^{w_i}\cap \theta^v$ within distance $r + 3M$, where $M$ is the bound on the width of a strip and length of a saddle connection from Lemma~\ref{L:strip-and-saddle-bound}. Since the path metric $d_{\theta^v}$
on $\theta^v$ is coarsely equivalent to the subspace metric, $d_{\theta^v}(p_1,p_2)$ is bounded in terms of $r$. The path metric on $\theta^v$ is biLipschitz equivalent to the $\ell^1$--metric on the product $\theta^v_{X_v}\times \mathbb R$. Since each edge of $\theta^v_{X_v}$ has definite length, there is a path from $p_1$ to $p_2$ in $\theta^v$ obtained by concatenating boundedly many (in terms of $r$) paths $\alpha_i$ with $f_{X_v}(\alpha_i)$ a saddle connection in $\theta^v_{X_v}$. Since each $\alpha_i$ is contained in some $\bTheta^{w}$, we are done.
\end{proof}

\begin{corollary}
\label{cor:inclusion_Lipschitz}
For any $x,y\in \partial \bTheta^v$, we have $d_{\bXi^v}(i^v(x), i^v(y))\le N(d_{\bTheta^v}(x,y))+1$.
\end{corollary}
\begin{proof}
Let $w = i^v(x)$ and $w' = i^v(y)$. Since $d_{\bTheta^v}$ and $\bar d$ are path metrics, we have $\bar d(x,y) \le d_{\bTheta^v}(x,y)$. By Lemma~\ref{lem:parameter_for_Xi}, $x$ and $y$ may be joined by a concatenation $\alpha_1\dotsb\alpha_k$ of $k \le N(d_{\bTheta^v}(x,y))+2$ paths $\alpha_j$ each of which lies in some $\bTheta^v\cap \bTheta^{w_j}$, and where $w = w_1$ and $w' =w_k$. For successive paths $\alpha_j,\alpha_{j+1}$, the vertices $w_j,w_{j+1}$ are adjacent in $\bXi^v$ by definition. Therefore $d_{\bXi^v}(w,w') \le k-1 \le N(d_{\bTheta^v}(x,y)) +1$.
\end{proof}

\begin{lemma}\label{lem:Xi_hyp}
Each $\bXi^v$ is uniformly quasi-isometric to a tree.  In particular, there exists $\delta > 0$ so that each $\bXi^v$ is $\delta$--hyperbolic
\end{lemma}

\begin{proof}
We appeal to Proposition \ref{prop:bottleneck} and show that for any vertices $w,w'$ of $\bXi^v$ there exist a path $\gamma(w,w')$ so that any path from $w$ to $w'$ passes within distance $3$ of each vertex of $\gamma(w,w')$.

First, note that $\bXi^v$ is isomorphic to the intersection graph of the collection of strips in $\bTheta_0^v$.  For each strip we have a vertex, and for each saddle connection of the spine $\theta_0^v$, there is an edge of $\bXi^v$ that connects the vertices corresponding to the strips that contain the saddle connection.  For each cone point in the spine $\theta_0^v$, there is also a complete graph on the vertices corresponding to strips that contain this cone point.  This accounts for all edges (because intersections of strips either arise along saddle connections or single cone points), and we note that the closure of each edge of the first type separates $\bXi^v$ into two components.

Suppose $w,w' \in \bXi^v$ are two vertices, and let $x,x' \in \theta_0^v$ be points in the (boundaries of the) strips corresponding to $w$ and $w'$, respectively, that are closest in $\bTheta_0^v$, and consider the geodesic in $\theta_0^v$ connecting these points, which is a concatenation of saddle connections $\sdl_1 \sdl_2 \cdots \sdl_n$.  For each $1 \leq i \leq n$, let $w_i^\pm$ be the vertices corresponding to the two strips $A_i^\pm$ that intersect in the saddle connection $\sdl_i$. We can form an edge path $\gamma(w,w')$ in $\bXi^v$, containing $w,w'$, and the $w_i^+$ as vertices, since $A_i^+ \cap A_{i+1}^+ \neq \emptyset$.
Observe that any path from $x$ to $x'$ must pass through the union $A_i^+ \cup A_i^-$, for each $i$, since $x$ and $x'$ lie in the closures of distinct components of $\bTheta_0^v \smallsetminus (A_i^+ \cup A_i^-)$.

Now let $w = w_0,w_1,\ldots,w_k = w'$ be the vertices of an edge path connecting $w$ to $w'$ in $\bXi^v$.  For any points in the strips corresponding to $w$ and $w'$, respectively, it is easy to construct a path in $\bTheta_0^v$ between these points that decomposes as a concatenation $\nu_1 \nu_2 \cdots \nu_k$ so that $\nu_j$ is contained entirely in the strip corresponding to $w_j$.  From the previous paragraph, this path must pass through $A_i^+ \cup A_i^-$, for each $i =1,\ldots,n$.  It follows that for each $1 \leq i \leq n$, the edge path must meet the union of the stars $star(w_i^+) \cup star(w_i^-)$.  Since these stars intersect, their union has diameter at most $3$, and we are done.
\end{proof}

\subsection{Windows and bridges}

For $v\in \vtx$, consider the set $\bar{\Sigma}^v$  of points in $\bar{\Sigma}$ that are inside some $v$--spine, as well as those points $\bar{\Sigma}^{\notin v}$ that are outside every $v$--spine:
\[ \bar{\Sigma}^v = \bigcup_{X \in \bar D} (\theta^v_X\cap \Sigma)\qquad\text{and}\qquad \bar{\Sigma}^{\not\in v} = \bar{\Sigma} \setminus \bar{\Sigma}^v.\]

For each $Y\in \bar D$ we now define a \emph{window map} $\Pi^v_Y \colon \bar{\Sigma} \to  \mathbb P(\partial \bTheta_Y^v)$ from cone points to the set of subsets of the boundary $\partial \bTheta_Y^v \subset \bTheta^v_Y$. There are two cases. Firstly,
\[\text{for $x\in \bar{\Sigma}^{\notin v}$,}\quad \Pi^v_Y(x) = \big\{ z \in \partial \bTheta^v_Y \mid [f_Y(x),z] \cap \bTheta^v_Y = \{z\} \big\}.\]
In words, $\Pi^v_Y(x)$ is the union of entrance points in $\bTheta^v_Y$ of any flat geodesic in $E_Y$ from $f_Y(x)$ to $\bTheta^v_Y$ (basically the closest point projection in $E_Y$), and we call it the \emph{window} for $x$ in $\bTheta^v_Y$.
Observe that for any $X,Y\in \bar D$ we have $f_{X,Y}(\Pi_Y^v(x)) = \Pi_X^v(x)$.
The second case, that of $\bar{\Sigma}^{v}$, is handled slightly differently:
\begin{align*}
\text{for $x\in \bar{\Sigma}^v$,}\quad \Pi^v_Y(x) &= f_{Y,X}(\Pi^v_X(x)), \quad\text{where}
\quad X = c_v(\pi(x))\in \partial B_v \\
\text{and}\quad \Pi^v_X(x) &= \big\{z \in \partial \bTheta^v_X \mid z  \text{ is a closest cone point to } f_X(x)\in \theta^v_X\big\}.
\end{align*}
Thus affine invariance $\Pi^v_Z(x) = f_{Z,Y}(\Pi^v_Y(x))$ is built directly into the definition.

Now for any $Y \in \bar D$ and $x,y \in \bar{\Sigma}$, we define
\[ d_Y^v(x,y) = \diam(\Pi^v_Y(x) \cup \Pi^v_Y(y)),\]
where the distance is computed in the path metric on $E_Y$ (or equivalently on $\bTheta^v_Y$).
Finally, we extend window maps to arbitrary subsets by declaring
\[\text{for $U\subset \bar E$,}\quad \Pi^v_Y(U) = \Pi^v_Y(U\cap \bar{\Sigma}) = \bigcup_{x\in \bar{\Sigma}\cap U} \Pi^v_Y(x)\]
which has the same effect as defining $\Pi^v_Y(x) = \emptyset \subset \partial \bTheta^v_Y$ for $x\notin \bar{\Sigma}$.

\begin{lemma} \label{lem:different spine}
If $x,y \in \bar{\Sigma}^{\not \in v}$ satisfy $f(x) = f(y)$, then $\Pi^v_Y(x) = \Pi^v_Y(y)$ for all $Y\in \bar D$.
\end{lemma}
\begin{proof}
This is immediate since $\Pi^v_Y(x)$ is defined just in terms of $f_Y(x) = f_Y(y)$.
\end{proof}

The following gives a counterpoint to Lemma~\ref{lem:different spine} for points in $\bar{\Sigma}^{v}$.

\begin{lemma} \label{lem:combo Lipschitz}
 There exists $K_2>0$ such that for any $v \in \vtx$ the following holds:  If $x, y \in \bar{\Sigma}^v$ satisfy $f(x) = f(y)$ and either
 \begin{enumerate}
 \item $x$ and $y$ are connected by a horizontal geodesic of length  $\leq 1$, or
 \item $x$ and $y$ are contained in $\partial \mathcal{B}_{w}$ for some $w \in \vtx$ with $\alpha(w) \neq \alpha(v)$,
 \end{enumerate}
 then $d_X^v(x,y) \leq K_2$, where $X = c_v(\pi(x))$.
 \end{lemma}
 \begin{proof}
Set $X = c_v(\pi(x))$ and $Y = c_v(\pi(y))$. Since $c_v\colon D \to \partial B_v$ is $1$--Lipschitz and $\diam(c_v(\partial B_w))$ is uniformly bounded for all such $w$, either condition (1) or (2) gives a uniform bound $K>0$ on the distance between $X$ and $Y$. Hence $f_{X,Y}$ is $e^{K}$--biLipschitz. The distance between $f_Y(y)\in \theta^v_Y$ and its closest cone points $\Pi^v_Y(y)$ in $\partial \bTheta^v_Y$ is also uniformly bounded by $2M$, by Lemma~\ref{L:strip-and-saddle-bound}. The same holds for the distance between $f_X(x)$ and $\Pi^v_X(x)$. It follows that $\Pi^v_X(y) = f_{X,Y}(\Pi^v_Y(y))$ lies within distance $2e^KM$ of $f_X(x) = f_{X,Y}(f_Y(y))$ and hence within distance $2e^KM  + 2M$ of $\Pi^v_X(x)$.
 \end{proof}

The next lemma explains that the image of $\Pi^v_Y$ is   not so far from being a point.

\begin{lemma}[Window lemma]\label{lem:window}
For any $v \in \vtx$, $Y\in \bar D$, and $x \in \bar{\Sigma}^{\not\in v}$, the window $\Pi^v_Y(x) \subset \partial \bTheta^v_Y$ is either a cone point or a single saddle connection.
\end{lemma}

\begin{proof}
If $f_Y(x)\in \partial \bTheta^v_Y$ then clearly $\Pi^v_Y(x)$ is the cone point $f_Y(x)$ itself. So suppose $f_Y(x)\notin \bTheta^v_y$ and let $\ell$ be the component of $\partial \bTheta^v_Y$ separating $f_Y(x)$ from $\theta^v_Y$ in $E_Y$, so that $\Pi^v_Y(x) \subset \ell$. Take any flat geodesic $[f_Y(x), z]$ in $E_Y$ from $f_Y(x)$ to a cone point $z\in \ell$.  The geodesic $[f_Y(x), z]$ first meets $\ell$ in some cone point $p$. Since the total cone angle at $p$ is at least $3\pi$ and the angle at $p$ along the side of $\ell$ containing $\bTheta^v$ is exactly $\pi$, the last saddle connection $\delta$ in the geodesic $[f_Y(x), p] \subset [f_Y(x),z]$ must make an angle of at least $\pi$ with $\ell$ on one side. Hence any geodesic from $f_Y(x)$ to a cone point on that side of $p$ must pass through $p$.

\begin{figure}
\label{fig:windows}
\begin{tikzpicture}[scale=1.5,inner sep=0pt, dot/.style={fill=black,circle,minimum size=4pt}]
\pgfmathsetmacro\sw{.5};
\pgfmathsetmacro\sh{4};
\coordinate (origin) at (0,-2);
\draw [fill,color=blue,opacity=.3] (origin) rectangle node[opacity=1,color=black,rotate=90] {\bf The strip between $\theta^v_Y$ and $f_Y(x)$} ++(\sw,\sh);
\draw[thick] (origin) -- ++(0,\sh);
\draw[thick] (origin) ++(\sw,0) -- ++(0,\sh);
\foreach \x in {.15, .3, .65, .9}
{

  \path (origin) ++(0,\x*\sh) node[dot] {};
  \draw[thick] (origin) ++(-1.2*\sw,\x*\sh) node[dot] {} -- ++(1.2*\sw,0);
  \draw[thick] (origin) ++(-1.2*\sw,\x*\sh) node[dot] {} -- ++(140:.8*\sw);
  \draw[thick] (origin) ++(-1.2*\sw,\x*\sh) node[dot] {} -- ++(235:.6*\sw);
}
\path (origin) ++(2.6*\sw,.6*\sh) coordinate (pt);
\path (pt) node[dot] {};
\draw[ultra thick,red] (origin) ++(\sw,.5*\sh) -- node (spot) {} ++(0,.2*\sh);
\path (origin) ++(-2*\sw,.5*\sh) node {The $v$ spine, $\theta^v_Y$};

\draw[thick,red,<-] (spot) -- ++(20:2.5*\sw) node[above right,color=black] {The window $\Pi^v_Y(x)$};
\draw[thick] (origin) ++(\sw,.5*\sh) node[below right] {\hspace{5pt}$\ge \pi$} -- (pt);
\draw[thick] (origin) ++(\sw,.7*\sh) node[above right] {\hspace{5pt}$\ge \pi$} -- (pt);
\foreach \x in {.1, .2, .5, .7, .85, 1}
{

  \path (origin) ++(\sw,\x*\sh) node[dot] {};
}
\draw[thick] (pt) -- ++(15:.8*\sw) node[dot] {}
                        -- ++(-12:\sw) node[dot] {}
                        -- ++(-45:\sw*1.2) node[dot] {}
                        -- ++(30:\sw) node[dot] {} node[below right] {$f_Y(x)$};
\end{tikzpicture}
\caption{A window} \label{F:window}
\end{figure}

If both angles between $\delta$ and $\ell$ at $p$ are at least $\pi$, then any geodesic from $f_Y(x)$ to $\ell$ passes through $p$. Hence $p$ is the unique point in $\partial \bTheta^v_Y$ closest to $f_Y(x)$ and $\Pi^v_Y(x) = \{p\}$ is a cone point as required.  Otherwise, consider the flat geodesic from $f_Y(x)$ to the adjacent cone point $p'$ on the other side of $p$ along $\ell$. The last saddle connection of this geodesic must also make an angle with $\ell$ of at least $\pi$ on one side.  This cannot be the side containing $p$, or else the geodesic from $f_Y(x)$ to $p$ would pass through $p'$ contradicting our choice of $p$. Hence any geodesic from $f_Y(x)$ to a cone point on the opposite side of $p'$ must pass through $p'$. Therefore $\Pi^v_Y(x)$ is the saddle connection between $p$ and $p'$, and we are done.  See Figure~\ref{F:window}.
\end{proof}

The following lemma gives us partial control over the window for points in adjacent vertex spaces in the same Bass--Serre tree.

\begin{lemma}[Bridge lemma] \label{lem:bridge}
For any $v, w \in \vtx$ with $d_{\tree}(v,w) = 1$, any $Y \in \bar D$, and any component $U \subset E_Y \smallsetminus \theta_Y^w$ not containing $\theta^v_Y$, there exists a (possibly degenerate) saddle connection $\delta_U \subset \partial \bTheta^v_Y$ with the following property: Every $x \in \bar{\Sigma}$ with $f_Y(x) \in \overline{U}$ satisfies $\Pi^v_Y(x) \subset \delta_U$.
 \end{lemma}

We call $\delta_U$ the {\em bridge} for $U$ in $E_Y$.  It is clear from the construction in the proof below that $f_{Z,Y}(\delta_U)$ is the bridge for $f_{Z,Y}(U)$, for any $Z \in \bar D$.

\begin{proof}
Let $U$ be as in the statement and $W$ be the component of $E_Y \smallsetminus \theta^w_Y$ containing $\theta^v_Y$.  Let $\gamma_U = \overline{U} \cap \theta^w_Y$ and $\gamma_W = \overline{W} \cap \theta^w_Y \subset \partial \bTheta^v_Y$, which are both bi-infinite flat geodesics in $\theta^w_Y$.

If $\gamma_U \cap \gamma_W = \emptyset$, then there is a unique geodesic between them in $\theta^w_Y$, and we take $\delta_U$ to be the endpoint of this geodesic which lies along $\gamma_W$.
On the other hand, if $\gamma_U \cap \gamma_W \neq \emptyset$, then their intersection is contained in the boundary of a strip along $\theta^v_Y$ and another along $\theta^w_Y$.  Two distinct strips in the same direction that intersect do so in either a single point or a single saddle connection, and hence $\gamma_U \cap \gamma_W$ is a point or single saddle connection, and we call $\delta_U$. See Figure~\ref{F:bridges}.

\begin{figure}[htb]
\label{fig:bridges}
\begin{tikzpicture}[scale=1.2,inner sep=0pt, dot/.style={fill=black,circle,minimum size=3pt}]
\coordinate (origin) at (0,0);

\pgfmathsetmacro\sw{.5};
\pgfmathsetmacro\sh{3};

\fill[color=blue,opacity=.3] (origin) rectangle ++(\sw,\sh);

\path (origin) ++(0,.5*\sh) coordinate (v) node[dot]{};
\draw[thick] (v) -- ++(-90:.4*\sh) coordinate (v1) node[dot]{};
\draw[thick] (v) -- ++(180:.8) coordinate (v2) node[dot]{};
\draw[thick] (v) -- ++(90:.35*\sh) coordinate (v3) node[dot]{};
\draw[thick] (v1) -- ++(-90:.1*\sh) coordinate (v11);
\draw[thick] (v1) -- ++(210:.5);
\draw[thick] (v3) -- ++(90:.15*\sh);
\draw[thick] (v3) -- ++(180:.5) coordinate (v31) node[dot]{};
\draw[thick] (v3) -- ++(180:1) coordinate (v32) node[dot]{};
\draw[thick] (v3) -- ++(180:1.5);
\draw[thick] (v31) -- ++(135:.6);
\draw[thick] (v32) -- ++(230:.6);
\draw[thick] (v2) -- ++(180:.5) coordinate (v21) node[dot]{};
\draw[thick] (v2) -- ++(180:1);
\draw[thick] (v21) -- ++(140:.5);
\draw[thick] (v2) -- ++(230:.5);

\path (origin) ++(\sw,.7*\sh) coordinate (w) node[dot]{};
\draw[thick] (w) -- ++(-90:.3*\sh) coordinate (w1) node[dot]{};
\draw[thick] (w) -- ++(-90:.5*\sh) coordinate (w2) node[dot]{};
\draw[thick] (w) -- ++(-90:.7*\sh);
\draw[thick] (w2) -- ++(-10:.7) coordinate (w21) node[dot]{};
\draw[thick] (w21) -- ++(-40:.7) coordinate (w211);
\draw[thick] (w1) -- ++(-5:1) coordinate (w11) node[dot]{};
\draw[thick] (w11) -- ++(-20:.8) coordinate (w111);
\draw[thick] (w11) -- ++(30:.8);
\draw[thick] (w) -- ++(90:.3*\sh);
\draw[thick] (w) -- ++(10:1.1) coordinate (w3) node[dot]{};
\draw[thick] (w3) -- ++(-30:1.1) coordinate (w31);
\draw[thick] (w3) -- ++(80:.7) coordinate (w32);

\filldraw[opacity=.2,color=teal] (w3) -- (w31) to[out=10,in=20] (w32) -- (w3);
\path (w21) ++(15:.6) node {$U'$};
\filldraw[opacity=.2,color=green] (w1) -- (w2) -- (w21) -- (w211) to[out=0,in=280] (w111) -- (w11) -- (w1);
\path (w3) ++(20:.5) node {$U$};

\draw[color=red,ultra thick] (w1) -- node[label={[label distance=1]0:{$\delta_{U'}$}}] {} (w2);
\filldraw[color=red] (w1) circle[radius=.06];
\filldraw[color=red] (w2) circle[radius=.06];
\filldraw[color=red] (w) circle[radius=.06] node[label={[label distance=2.8]65:{$\delta_{U}$}}] {};

\path (w1) ++(35:.8) node {$\theta_Y^{w}$};
\path (v) ++(235:1) node {$\theta^v_Y$};
\end{tikzpicture}

\caption{Bridges, and the proof of Lemma \ref{lem:bridge}.} \label{F:bridges}
\end{figure}

Now consider any point $x \in \bar{\Sigma}$ with $f_Y(x) \in \overline{U}$. Observe that $x\in \bar{\Sigma}^{\notin v}$ so that $\Pi^v_Y(x)$ falls under the first case of the window definition.
Further, any flat geodesic from $f_Y(x)$ to $\bTheta^v_Y$ must pass through both $\gamma_U$ and $\gamma_W$, and hence must pass through $\delta_U \subset \partial \bTheta^v_Y$.  It follows that $\Pi^v_Y(x) \subset \delta_U$, as required.
\end{proof}

The following is an easy consequence of the previous lemma.

\begin{corollary}\label{cor:bridge same}
For any $v, w \in \vtx$ with $2 \leq d_{\tree}(v,w) < \infty$ and $Y \in \bar D$, there exists a (possibly degenerate) saddle connection $\delta^v_Y(w) \subset \partial \bTheta^v_Y$ so that if $x \in \bar{\Sigma}$ has $f_Y(x) \in \bTheta^w_Y$, then $\Pi^v_Y(x) \subset \delta^v_Y(w)$. In particular, $\Pi^v_Y(\bTheta^w)\subset \delta^v_Y(w)$.
\end{corollary}

\begin{proof}
There exists $u \in T_{\alpha(v)}^{(0)}$ between $v$ and $w$ with $d_{\tree}(u,v) = 1$, and a component $U  \subset E_Y \setminus \theta^u_Y$ whose closure contains $\bTheta^w_Y$.  Setting $\delta^v_Y(w) = \delta_U$ and applying Lemma \ref{lem:bridge} completes the proof.
\end{proof}

The next corollary is similar.

\begin{corollary}\label{cor:bridge diff}
If $v, w \in \vtx$ satisfy $d_{\tree}(v,w) = \infty$ and $\theta^w_0 \cap \theta^v_0 = \emptyset$, then for each $Y\in \bar D$ there is a connected union $\delta^v_Y(w) \subset \partial \bTheta^v_Y$ of at most two saddle connections such that $\Pi^v_Y(x) \subset \delta^v_Y(w)$ for all $x\in \bar{\Sigma}^w$. In particular $\Pi^v_Y(\theta^w)\subset \delta^v_Y(w)$.
\end{corollary}

\begin{proof}
From the hypotheses, $\theta^w_Y$ is contained in some component $W \subset E_Y \smallsetminus \theta^v_Y$.   Let $u \in \vtx$ be such that $d_{\tree}(v,u) = 1$ and $\theta^u_Y \subset W$.  Since $\alpha(u) = \alpha(v) \neq \alpha(w)$,  $\theta^u_Y$ and $\theta^w_Y$ can intersect in at most one point. If $\theta^u_Y\cap \theta^w_Y= \emptyset$, then $\theta^w_Y$ is contained in a component $U$ of $E_Y\setminus \theta^u_Y$ disjoint from $\theta^v_Y$; thus $\Pi^v_Y(\theta^w_Y)$ is contained in the bridge $\delta_U$ for $U$ by Lemma~\ref{lem:bridge}.
Otherwise $\theta^u_Y\cap \theta^w_Y \ne \emptyset$, and we claim there is a cone point $p\in \theta^u_Y$ such that $p\in \overline{U}$ for every component $U$ of $E_Y\setminus \theta^u_Y$ that intersects  $\theta^w_Y$. Indeed, if $\theta^u_Y\cap \theta^w_Y$ is a cone point, we take $p$ to be this intersection point, and if not  $\theta^u_Y\cap \theta^w_Y$ is an interior point of a saddle connection of $\theta^u_Y$ and we may take $p$ to be either of its endpoints.
For each component $U \subset E_Y\setminus \theta^u_Y$ that intersects $\theta^w_Y$ we then have $\Pi^v_Y(p) \subset \delta_U$, where $\delta_U$ is the bridge for $U$.  Since $f_Y(\bar{\Sigma}^w)\subset \theta^w_Y$ is contained in the union of the closures of such $U$, it follows that $\Pi^v_Y(\bar{\Sigma}^w)$ is contained in a union of saddle connections along $\partial \bTheta^v_Y$, all of which contain $\Pi^v_Y(p)$, and hence is a connected union of at most two saddle connections.  This completes the proof.
\end{proof}

The final case to consider is that of spines in different directions that intersect:

\begin{lemma}\label{lem:crossing spines}
There exists $K_3>0$ such that if $v, w \in \vtx$ with $d_{\tree}(v,w) = \infty$ and $\theta^w_0 \cap \theta^v_0 \neq \emptyset$, then $\diam (\Pi^v_Y(\theta^w))  < K_3$ for all $Y \in c_v(\partial B_w)$.
 \end{lemma}

\begin{figure}[htb]
\usetikzlibrary{decorations.pathreplacing}
\begin{tikzpicture}[scale=1.8,inner sep=0pt, dot/.style={fill=black,circle,minimum size=3pt}]

\pgfmathsetmacro\spokes{5};
\pgfmathsetmacro\spokeslessone{\spokes-1};
\pgfmathsetmacro\angle{360/\spokes};
\pgfmathsetmacro\rad{1.8};
\pgfmathsetmacro\inrad{.5*\rad};
\pgfmathsetmacro\SpineAngleOffset{\angle/10};
\pgfmathsetmacro\WindowAngleOffset{\angle/4};

\coordinate (origin) at (0,0);
\path (origin) node[dot]{};
\foreach \i in {1,...,\spokes}
{
  \path (origin) ++(\i*\angle:\rad) coordinate (v\i);
  \draw[thick,dashed] (origin) -- (v\i);

  \path (origin) ++(\i*\angle+\SpineAngleOffset:\rad) coordinate (sr\i);
  \path (origin) ++(\i*\angle+\angle-\SpineAngleOffset:\rad) coordinate (sl\i);
  \path (origin) ++(\i*\angle+.5*\angle-\WindowAngleOffset:\inrad) coordinate (mr\i);
  \path (origin) ++(\i*\angle+.5*\angle+\WindowAngleOffset:\inrad) coordinate (ml\i);
  \ifthenelse{\isodd{\i}}
  {
    \pgfmathsetmacro\dist{.37*\rad}
  }{
    \pgfmathsetmacro\dist{.65*\rad}
  };
  \path (origin) ++(\i*\angle+.5*\angle:\dist) coordinate (w\i);
  \path (origin) ++(\i*\angle+.33*\angle:\rad) coordinate (b\i);
  \path (origin) ++(\i*\angle+.66*\angle:\rad) coordinate (c\i);

  \draw[ultra thick, blue!60!black] (origin) -- (w\i);
  \draw[thick] (b\i) -- (w\i) -- (c\i);
  \ifthenelse{\isodd{\i}}
  {
    \path (w\i) coordinate (m\i) node[dot,red]{};
  }
  {
    \path (mr\i) -- coordinate[midway](m\i) (ml\i);

    \filldraw[color=blue!60!black] (w\i) circle[radius=0.04];
  }

  \fill[color=blue,opacity=.2] (origin) ++(\i*\angle+\angle:\rad) -- (origin) -- (v\i) -- (sr\i) -- (mr\i) -- (m\i) -- (ml\i) -- (sl\i) -- cycle;
  \draw[thick] (sr\i) -- (mr\i) (ml\i) -- (sl\i);
  \draw[ultra thick, red] (mr\i) -- (m\i) -- (ml\i);
  \filldraw[color=red] (mr\i) circle[radius=.04] (ml\i) circle[radius=.04];
  \ifthenelse{\isodd{\i}}{\filldraw[color=red] (m\i) circle[radius=0.04];}{}
}

  \path (b\spokes) node[above right] {$\theta^w_Y$};
  \path (origin) ++(\spokes*\angle+.69*\angle-\angle:.32*\rad) node[color=blue!60!black] {$W_0$};
  \draw [decorate,decoration={brace,amplitude=5pt,mirror,raise=4pt},yshift=0pt] (sl\spokeslessone) -- (sr\spokes) node [black,midway,xshift=0.8cm] {\footnotesize $\bTheta^v_Y$};

\end{tikzpicture}
\caption{The proof of Lemma~\ref{lem:crossing spines}}
\label{fig:for_crossing_spines}
\end{figure}

\begin{proof} Let $x_0 = \theta^v_Y \cap \theta^w_Y$ be the unique intersection point of the spines.  Let $W_0$ be the smallest subgraph of $\theta^w_Y$ containing $\theta^w_Y \cap \bTheta^v_Y$ and let $W_1, \dots, W_k$ be the closures of the components of $\theta^w_Y \setminus W_0$, so that $\theta^w_Y = W_0 \cup \dots \cup W_k$. See Figure~\ref{fig:for_crossing_spines}. For $1 \leq i \leq k$, let  $x_i \in W_i$ be the closest (cone) point to $x_0$. Then define $p_i$ to be the intersection of the geodesic $[x_0, x_i]\subset\theta^w_Y$ with $\partial\bTheta^v_Y$ and let $\delta_i\subset \partial \bTheta^v_Y$ be the segment consisting of the ($1$ or $2$) saddle connections along $\partial \bTheta^v_Y$ that contain $p_i$. Define $\delta^v_Y(w) = \delta_1\cup \dots \cup \delta_k$, and note that $\delta^v_X(w) = f_{X,Y}(\delta^v_Y(w))$ and $\theta^w_Y\cap \partial \bTheta^v_Y\subset \delta^v_Y(w)$ by construction.

For any $x\in \bar{\Sigma}$ with $f_Y(x) \in W_i$, where $i =1, \dots, k$, the flat geodesic from $f_Y(x)$ to $x_0$ first intersects $\bTheta^v_Y$ at $p_i$; therefore $p_i\in \Pi^v_Y(x)$ by definition of the window. By Lemma~\ref{lem:window}, it follows that $\Pi^v_Y(x)\subset \delta_i$. The union $\cup_{i=1}^k W_i$ contains every cone point of $\theta^w_Y$ except possibly $x_0$. Thus we have proven $\Pi^v_Y(\theta^w\cap \bar{\Sigma}^{\notin v}) = \Pi^v_Y(\theta^w_Y\cap \bar{\Sigma}^{\notin v}) \subset \delta^v_Y(w)$.

Let $Z\in D$ be the closest point on $\partial B_v$ to $\partial B_w$, thus $Z\in \partial B_v$ lies on the unique hyperbolic geodesic that intersects $\partial B_v$ and $\partial B_w$ orthogonally. In $E_Z$ the directions $\alpha(v)$ and $\alpha(w)$ are perpendicular. Therefore the cone points of $\partial \bTheta^v_Z$ that are closest to $\theta^v_Z\cap \theta^w_Z$ all lie in $\delta^v_Z(w)$, since they must be endpoints of saddle connections along $\partial \bTheta^v_Z$ that intersect $\theta^w_Z$.
More generally, for any $Y\in c_v(\partial B_w)$, the directions $\alpha(v)$ and $\alpha(w)$ are nearly perpendicular and thus we have
\begin{equation*}
\label{eqn:perpendicular-crossing-spines}
\Pi^v_Y(\theta^w\cap \bar{\Sigma}^{\notin v}) \subset \delta^v_Y(w) \subset B_K(\theta^v_Y\cap \theta^w_Y).
\end{equation*}
for some uniform constant $K > 0$ that depends only on the length of $c_v(\partial B_w)$ and the maximum over $\partial B_v$ of the length/width of any saddle connection/strip in the $\alpha(v)$ direction (Lemma~\ref{L:strip-and-saddle-bound}).
Now, for any $x\in \theta^w \cap \bar{\Sigma}^v$,  the point $X = c_v(\pi(x))$ lies in $c_v(\partial B_w)$ and we have that $f_X(x) = \theta^v_X\cap \theta^w_X$. The above equation shows there are cone points of $\partial \bTheta^v_X$ within $K$ of $f_X(x)$; hence $\Pi^v_X(x)=\Pi^v_X(f_X(x))$ lies in $B_K(\theta^v_X\cap \theta^w_X)$ by definition.
Using the fact that the length of $c_v(\partial B_w)$ is uniformly bounded, we see that the map $f_{Y,X}$ is uniformly biLipschitz and therefore that $\Pi^v_Y(x) = f_{Y,X}(\Pi^v_X(x))$ lies within bounded distance of $\theta^v_Y\cap \theta^w_Y = f_{Y,X}(\theta^v_X\cap \theta^w_X)$. Combining this the above finding that $\Pi^v_Y(\theta^w\cap \bar{\Sigma}^{\notin v})$ lies within bounded distance of $\theta^v_Y\cap \theta^w_Y$, we finally conclude that $\diam(\Pi^v_Y(\theta^w))$ is uniformly bounded.
\end{proof}

\subsection{Projections}
Here we define projections $\Lambda^v \colon \bar{\Sigma} \to \mathbb P(\mathcal K^v)$ and $\xi^v \colon \bar{\Sigma} \to \mathbb P(\bXi^v)$.
 In preparation, we first define $\Pi^v \colon \bar{\Sigma} \to \mathbb P(\partial \bTheta^v)$ by
\[ \Pi^v(x) = \bigcup_{Y \in \partial B_v}\Pi^v_Y(x),\quad\text{for any $v\in \vtx$ and $x\in \bar{\Sigma}$}.\]
In words, $\Pi^v(x)$ is consists of the $v$--windows of $x$ in all fibers over $\partial B_v$. As before, we extend to arbitrary subsets $U\subset \bar E$ by setting $\Pi^v(U) = \Pi^v(U\cap \bar{\Sigma})$.

Now, for each $v\in \vtx$ our projections are defined as the compositions
\[ \Lambda^v  = \lambda^v\circ \Pi^v \quad \mbox{ and } \quad \xi^v(x) = i^v\circ \Pi^v.\]
A useful observation is that for any two vertices $v,w \in \vtx$ with $d_\tree(v,w) = 1$ and $X \in \bar D$, we have
\begin{equation} \label{E:an obvious but useful fact}
\xi^v(\theta^w_X) = \xi^v(\theta^w)=w \in \bXi^v.
\end{equation}
Mirroring the notation $d_Y^v(x,y)$ above, for $x,y\in \bar{\Sigma}$ we define
\[d_{\mathcal{K}^v}(x,y) = \diam(\Lambda^v(x)\cup \Lambda^v(y))\quad\text{and}\quad d_{\bXi^v}(x,y) = \diam(\xi^v(x)\cup \xi^v(y)).\]

\begin{lemma} \label{L:bounded is bounded}
There is a function $N'\colon [0,\infty)\to [0,\infty)$ such that for all $v\in \vtx$, $X\in \partial B_v$, and $x,y\in \bar{\Sigma}$ the quantities $d_{\mathcal{K}^v}(x,y)$ and $d_{\bXi^v}(x,y)$ are at most $N'(d_X^v(x,y))$.
\end{lemma}
\begin{proof}
For $Y\in \partial B_v$, let us compare the images of some subset $U \subset \partial \bTheta^v_Y$ and
\[\mathrm{Smear}(U) = \bigcup_{Z\in \partial B_v} f_{Z,Y}(U) \subset \partial \bTheta^v\]
under the maps $\lambda^v,i^v$.
Since the boundary components of $\bTheta^v$ are preserved by the maps $f_{Z,Y}$, the images $i^v(U)=i^v(\mathrm{Smear}(U))$ are exactly the same.
Moreover, for each $x\in \partial \bTheta^v_Y$ we have $\lambda^v(\mathrm{Smear}(\{x\})) = \lambda^v(\ell_{x,\alpha(v)})$, and so by Proposition~\ref{prop:KL}(\ref{item:lambda_flowlines_bounded}) this is a single point.
Therefore $\lambda^v(U) = \lambda^v(\mathrm{Smear}(U))$.

Now, let $x,y\in \bar{\Sigma}$ and $X\in \partial B_v$ be as in the statement. Set $U = \Pi^v_X(x)\cup \Pi^v_X(y)$ so that  $\diam(U) = d^v_X(x,y)$. Since $\Pi^v(x) = \mathrm{Smear}(\Pi^v_X(x))$ and similarly for $\Pi^v(y)$, we see that
\[d_{\mathcal{K}^v}(x,y) = \diam(\lambda^v(\mathrm{Smear}(U))) \quad\text{and}\quad d_{\bXi^v}(x,y) = \diam(i^v(\mathrm{Smear}(U))).\]
Since $\lambda^v$ is coarsely Lipschitz by Proposition~\ref{prop:KL}, the preceding paragraph shows that $d_{\mathcal{K}^v}(x,y) = \diam(\lambda^v(U)) \leq K_1 d_X^v(x,y)$. Similarly, by Corollary~\ref{cor:inclusion_Lipschitz} we have that $\diam_{\bXi^v}(x,y) = \diam(i^v(U))\le N(d_X^v(x,y))+1$.
\end{proof}

 \begin{proposition} \label{prop:R-projections}
 There exists $K_4>0$ so that for any $v \in \vtx$:
 \begin{enumerate}
 \item $\Lambda^v$ and $\xi^v$ are $K_4$--coarsely Lipschitz;
 \item For any $w \in \vtx$, we have
 \begin{enumerate}
 \item $\diam \left(\Lambda^{v}(\bTheta^w)\right) < K_4$, unless $d_{\tree}(v,w) \leq 1$;
 \item $\diam \left(\xi^v(\bTheta^w)\right) < K_4$ unless $w = v$.
 \end{enumerate}
 \end{enumerate}
 \end{proposition}

 \begin{proof}
By Lemma~\ref{L:combinatorial paths in bar E}, any pair of points $x,y\in \bar{\Sigma}$ may be connected by a path of length at most $R \bar d(x,y)$ that is a concatenation of at most $R\bar d(x,y)+1$ pieces, each of which is either a saddle connection of length at most $R$ in a vertical vertical fiber, or a horizontal geodesic segment in $\bar E$.
By the triangle inequality, it thus suffices to assume that $x$ and $y$ are the endpoints of either a horizontal geodesic or a vertical saddle connection of length at most $R$. Appealing to Lemma~\ref{L:bounded is bounded}, it further suffices to show that $d_X^v(x,y)$ is linearly bounded by $\bar{d}(x,y)$ for some $X\in \partial B_v$. Lemmas \ref{lem:different spine} and \ref{lem:combo Lipschitz}(1) handle the horizontal segment case, since we are free to subdivide such a path into $\lceil\bar{d}(x,y)\rceil$ segments of length at most $1$.

For the vertical segment case we assume $x$ and $y$ lie in the same fiber $E_Y$ and differ by a saddle connection $\delta$ of length at most $R$. Let $\theta^w_Y$, where $w\in \vtx$, be the spine containing $\delta$. The fact that $\delta$ is bounded means that $Y = \pi(x)=\pi(y)$ is bounded distance from the horocycle $\partial B_w$. Let $X = c_w(Y)\in \partial B_w$ and let $\delta' = f_{X,Y}(\delta)$ be the saddle connection in $\theta^w_X$ connecting $x'=f_{X,Y}(x)$ and $y' = f_{X,Y}(y)$. By the triangle inequality and the first part above about bounded length horizontal segments, it suffices to work with the points $x',y'\in \theta^w$.  There are three cases to consider: Firstly, if $v = w$, then $x',y'\in \theta^v_X$ so that $\Pi^v_X(x')$ and, $\Pi^v_X(y')$ choose the closest cone points in $\partial \bTheta^v_X$ to $x'$ and $y'$, respectively. Since $x'$ and $y'$ are close, so are $\Pi^v_X(x')$ and $\Pi^v_X(y')$. Secondly, if $d_\tree(v,w) > 1$, then Corollaries \ref{cor:bridge same} and \ref{cor:bridge diff}, and Lemma \ref{lem:crossing spines} give a uniform bound on $d^v_Z(x',y') \le \diam(\Pi^v_Z(\theta^w))$ for any point $Z\in c_v(\partial B_w)$. Finally, if $d_\tree(v,w) = 1$ then $\theta^w_X$ and $\theta^v_X$ are adjacent non-crossing spines in $E_X$. Since $\theta^w_X$ is totally geodesic, if follows that $\Pi^v_X(x')$ and $\Pi^v_X(y')$ are either equal or connected by a single edge of $\partial \bTheta^v_X$. But this saddle connection has uniformly bounded length,  since $X\in \partial B_v$, which completes the proof of (1).

For (2), first recall that strips/saddle connections in the $\alpha(w)$ direction have uniformly bounded width/length over $\partial B_w$ (Lemma~\ref{L:strip-and-saddle-bound}). Therefore $\bTheta^w\cap \bar{\Sigma}$ is contained in a bounded neighborhood of $\theta^w\cap \bar{\Sigma}$. By part (1) it thus suffices to bound $\diam(\Lambda^v(\theta^w))$ and $\diam(\xi^v(\theta^w))$. When $d_{\tree}(v,w) \ge 2$, Corollaries \ref{cor:bridge same} and \ref{cor:bridge diff}, and Lemma \ref{lem:crossing spines}, imply that there exists $X \in \partial B_v$ so that $\Pi^v_X(\theta^w)$ has bounded diameter in $\bTheta^v_X$. Appealing to Lemma~\ref{L:bounded is bounded} now bounds $\diam(\Lambda^v(\theta^w))$ and $\diam(\xi^v(\theta^w))$ in these cases. For the remaining case $d_\tree(v,w)=1$ of 2(b), we note that $\xi^v(\theta^w)$ is a single point by \eqref{E:an obvious but useful fact} and 2(b) follows.
\end{proof}

\section{Hierarchical hyperbolicity of $\Gamma$} \label{S:combinatorial HHS}

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In this section we complete the proof that $\Gamma$ is hierarchically hyperbolic. We will use a criterion from \cite{comb_HHS}, which we now briefly discuss.
For further information and heuristic discussion of this approach to hierarchical hyperbolicity, we refer the reader to \cite[\S1.5, ``User's guide and a simple example'']{comb_HHS}.

Consider a simplicial complex $\CX$ and a graph $\CW$ whose vertex set is the set of maximal simplices of $\CX$.
The pair $(\CX,\CW)$ is called a \emph{combinatorial HHS} if it satisfies the requirements listed in Definition~\ref{defn:combinatorial_HHS} below, and \cite[Theorem 1.18]{comb_HHS} guarantees that in this case $\CW$ is an HHS. The main requirement is along the lines of: $\CX$ is hyperbolic, and links of simplices of $\CX$ are also hyperbolic. However, this is rarely the case because co-dimension--1 faces of maximal simplices have discrete links. To rectify this, additional edges (coming from $\CW$) should be added to $\CX$ and its links as detailed in Definition~\ref{defn:X_graph}.
In our case, after adding these edges, $\CX$ will be quasi-isometric to $\hat E$, and each other link will be quasi-isometric to either a point or to one of the spaces $\mathcal K^v$ or $\bXi^v$  introduced in \S\ref{S:projections}.

There are two natural situations where such pairs arise that the reader might want to keep in mind. First, consider a group $H$ acting on a simplicial complex $\CX$ so that there is one orbit of maximal simplices, and those have trivial stabilizers. In this case, we take $\CW$ to be (a graph isomorphic to) a Cayley graph of $H$. (More generally, if the action is cocompact with finite stabilizers of maximal simplices, then the appropriate $\CW$ is quasi-isometric to a Cayley graph.) For the second situation, $\CX$ is the curve graph of a surface; then maximal simplices are pants decompositions of the surface and $\CW$ can be taken to be the pants graph. We will use this as a working example below, when we get into the details.

Most of the work carried out in \S\ref{S:projections} will be used (as a black-box) to prove that, roughly, links are quasi-isometrically embedded in a space obtained by removing all the ``obvious'' vertices that provide shortcuts between vertices of the link. This can be seen as an analogue of Bowditch's fineness condition in the context of relative hyperbolicity.

This section is organized as follows. In \S\ref{subsec:setup} we list all the relevant definitions and results from \cite{comb_HHS}, and we illustrate them using pants graphs. In \S\ref{subsec:our_comb_HHS} we construct the relevant combinatorial HHS for our purposes. In \S\ref{subsec:links_list} we analyze all the various links and related combinatorial objects;
we note that most of the work done in \S\ref{S:projections} is used here to prove Lemma \ref{lem:retractions}.
At that point, essentially only one property of combinatorial HHSs will be left to be checked, and we do so in \S\ref{subsec:final_proof}.

\subsection{Basic definitions}\label{subsec:setup}

We start by recalling some basic combinatorial definitions and constructions. Let $X$ be a flag simplicial
complex.

\begin{defn}[Join, link, star]\label{defn:join_link_star}
Given disjoint simplices $\Delta,\Delta'$ of $X$, the \emph{join} is denoted $\Delta\star\Delta'$ and is the simplex spanned by
$\Delta^{(0)}\cup\Delta'^{(0)}$, if it exists.  More generally, if $K,L$ are disjoint induced subcomplexes of
$X$ such that every vertex of $K$ is adjacent to every vertex of $L$, then the join
$K\star L$ is the induced subcomplex with vertex set $K^{(0)}\cup L^{(0)}$.

For each simplex $\Delta$, the \emph{link} $\link(\Delta)$ is the union of
all simplices $\Delta'$ of $X$ such that $\Delta'\cap\Delta=\emptyset$ and $\Delta'\star\Delta$ is a simplex of $X$.
The \emph{star} of $\Delta$ is $Star(\Delta)=\link(\Delta)\star\Delta$, i.e. the union of all simplices of $X$ that contain
$\Delta$.

We emphasize that $\emptyset$ is a simplex of $X$, whose link is all of $X$ and whose star is all of $X$.
\end{defn}

\begin{defn}[$X$--graph, $W$--augmented dual complex]\label{defn:X_graph}
An \emph{$X$--graph} is any graph $W$
whose vertex set is the set of maximal simplices of $X$ (those not contained in any larger simplex).

For a flag complex $X$ and an $X$--graph $W$, the \emph{$W$--augmented dual graph}
$\duaug{X}{W}$ is the graph defined as follows:
\begin{itemize}
     \item the $0$--skeleton of $\duaug{X}{W}$ is $X^{(0)}$;
     \item if $v,w\in X^{(0)}$ are adjacent in $X$, then they are adjacent in $\duaug{X}{W}$;
	 \item if two vertices in $W$ are adjacent, then we consider
	 $\sigma,\rho$, the associated maximal simplices of $X$, and
	 in $\duaug{X}{W}$ we connect each vertex of $\sigma$ to each vertex
	 of $\rho$.
\end{itemize}
We equip $W$ with the usual path-metric, in which each edge has unit length, and do the same for $\duaug{X}{W}$.  Observe that the $1$--skeleton of $X$ is a subgraph $X^{(1)} \subset \duaug{X}{W}$.
\end{defn}

We provide a running example to illustrate the various definitions in a familiar situation.  This example will not be used in the sequel.
\begin{example} \label{Ex:curve/pants graph 1} If $\X$ is the curve complex of the surface $S$, then an example of the an $X$--graph, $W$, is the pants graph, since a maximal simplex is precisely a pants decomposition.  The $W$--augmented dual graph can be thought of as adding to the curve graph, $X^{(0)}$ and edge between any two curves that fill a one-holed torus or four-holed sphere and intersect once or twice, respectively: indeed, these subsurfaces are precisely those where an {\em elementary move} happens as in the definition of adjacency in the pants graph.
\end{example}

\begin{defn}[Equivalent simplices, saturation]\label{defn:simplex_equivalence}
For $\Delta,\Delta'$ simplices of $X$, we write $\Delta\sim\Delta'$ to mean
$\link(\Delta)=\link(\Delta')$. We denote by $[\Delta]$ the equivalence class of $\Delta$.  Let $\sat(\Delta)$ denote
the set of vertices $v\in X$ for which there exists a simplex $\Delta'$ of $X$ such that $v\in\Delta'$ and
$\Delta'\sim\Delta$, i.e. $$\sat(\Delta)=\left(\bigcup_{\Delta'\in[\Delta]}\Delta'\right)^{(0)}.$$
We denote by $\mathfrak S$ the set of $\sim$--classes of non-maximal simplices in $X$.
\end{defn}

\begin{defn}[Complement, link subgraph]\label{defn:complement}
Let $W$ be an $X$--graph.  For each simplex $\Delta$ of $X$, let
$Y_\Delta$ be the subgraph of $\duaug{X}{W}$ induced by the set
of vertices $(\duaug{X}{W})^{(0)}-\sat(\Delta)$.

Let $\mathcal C(\Delta)$ be the full subgraph of $Y_\Delta$ spanned by $\link(\Delta)^{(0)}$.  Note that $\mathcal
C(\Delta)=\mathcal C(\Delta')$ whenever $\Delta\sim\Delta'$.  (We emphasize
that we are taking links in $X$, not in $X^{+W}$, and then considering the
subgraphs of $Y_\Delta$ induced by those links.)
\end{defn}

We now pause and continue with the illustrative example.
\begin{example} \label{Ex:curve/pants graph 2} Let $X$ and $W$ be as in Example~\ref{Ex:curve/pants graph 1}.  A simplex $\Delta$ is a multicurve which determines two (open) subsurfaces $U = U(\Delta),U' = U'(\Delta) \subset S$, where $U$ is the union of the complementary components of the multicurve that are not a pair of pants, and $U' = S - \overline{U}$.  Note that $\partial U \subset \Delta$ is a submulticurve and that $\Delta - \partial U$ is a pants decomposition of $U'$.  A simplex $\Delta'$ is equivalent to $\Delta$ if it defines the same subsurfaces.  Thus $\sat(\Delta)$ consists of $\partial U(\Delta)$ together with all essential curves in $U'(\Delta)$, while $\mathcal C(\Delta)$ is the join of the curve graphs of the components of $U(\Delta)$.  We note that this is true even when some components are one-holed tori or four-holed spheres (because of the extra edges in $\duaug{X}{W}$).
\end{example}

\begin{defn}[Nesting]\label{defn:nest}
 Let $X$ be a simplicial complex.  Let $\Delta,\Delta'$ be non-maximal simplices of $X$.  Then we write $[\Delta]\nest[\Delta']$ if $\link(\Delta)\subseteq\link(\Delta')$.
\end{defn}
We note that if $\Delta' \subset \Delta$, then $[\Delta] \nest [\Delta']$.  Also, for Example~\ref{Ex:curve/pants graph 1},\ref{Ex:curve/pants graph 2}, $[\Delta] \nest [\Delta']$ if and only if $U(\Delta) \subset U(\Delta')$.

Finally, we are ready for the main definition:

\begin{defn}[Combinatorial HHS]\label{defn:combinatorial_HHS}
 A \emph{combinatorial HHS} $(X,W)$ consists of a flag simplicial
 complex $X$ and an $X$--graph $W$ satisfying the following conditions for some $n \in \mathbb N$ and $\delta \geq 1$:
 \begin{enumerate}
  \item \label{item:chhs_flag}any chain $[\Delta_1]\propnest[\Delta_2]\propnest\dots$ has length at most $n$;
    \item \label{item:chhs_delta} for each non-maximal simplex $\Delta$, the subgraph
$\mathcal C(\Delta)$ is
$\delta$--hyperbolic and $(\delta,\delta)$--quasi-isometrically embedded in
$Y_\Delta$;
\item \label{item:chhs_join}Whenever $\Delta$ and $\Delta'$ are non-maximal simplices for which there exists
a non-maximal simplex $\Gamma$ such that $[\Gamma]\nest[\Delta]$, $[\Gamma]\nest[\Delta']$, and $\diam(\mathcal C
(\Gamma))\geq \delta$, then there exists a
simplex $\Pi$ in the link of $\Delta'$ such that $[\Delta'\star\Pi]\nest [\Delta]$ and all $[\Gamma]$ as above satisfy $[\Gamma]\nest[\Delta'\star\Pi]$;
\item \label{item:C_0=C} if $v,w$ are distinct non-adjacent vertices of $\link(\Delta)$, for some simplex $\Delta$ of $X$, contained in
$W$--adjacent maximal simplices, then they are contained in $W$--adjacent simplices of the form $\Delta\star\Delta'$.
\end{enumerate}
\end{defn}

We will see below that combinatorial HHS give HHSs. The reader not interested in the explicit description of the HHS structure can skip the following two definitions.

\begin{defn}[Orthogonality, transversality]\label{defn:orth}
Let $X$ be a simplicial complex.  Let $\Delta,\Delta'$ be non-maximal simplices of $X$.  Then we write $[\Delta]\orth[\Delta']$ if $\link(\Delta')\subseteq \link(\link(\Delta))$.
If $[\Delta]$ and $[\Delta']$ are not $\orth$--related or $\nest$--related, we write $[\Delta]\transverse[\Delta']$.
\end{defn}

\begin{defn}[Projections]\label{defn:projections}
Let $(X,W)$ be a combinatorial HHS.

Fix $[\Delta]\in\mathfrak S$ and define a map $\pi_{[\Delta]}\colon W\to \mathbb{P}({\mathcal C([\Delta])})$ as
follows.  First let $p\colon Y_\Delta\to \mathbb{P}({\mathcal C([\Delta])})$ be the coarse closest-point projection, i.e.
$$p(x)= \big\{y\in\mathcal C([\Delta]) \mid d_{Y_\Delta}(x,y)\leq d_{Y_\Delta}(x,\mathcal C([\Delta]))+1\big\}.$$

Suppose that $w\in W^{(0)}$, so $w$ corresponds to a unique simplex $\Delta_w$ of $X$.
 Define $$\pi_{[\Delta]}(w)=p(\Delta_w\cap Y_\Delta).$$

We have thus defined  $\pi_{[\Delta]}\colon W^{(0)}\to \mathbb{P}({\mathcal C([\Delta])})$.   If $v,w\in W^{(0)}$ are joined by an edge $e$ of $W$,
then $\Delta_v,\Delta_w$ are joined by edges in $\duaug{X}{W}$, and we let
$\pi_{[\Delta]}(e)=\pi_{[\Delta]}(v)\cup\pi_{[\Delta]}(w)$.

Now let $[\Delta],[\Delta']\in\mathfrak S$ satisfy $[\Delta]\transverse[\Delta']$ or $[\Delta']\propnest [\Delta]$.
Let $$\rho^{[\Delta']}_{[\Delta]}=p(
\sat(\Delta')\cap Y_\Delta).$$

Let $[\Delta]\propnest
[\Delta']$. Let $\rho^{[\Delta']}_{[\Delta]}\colon \mathcal C([\Delta'])\to
\mathcal C([\Delta])$ be the restriction of $p$ to $\mathcal C([\Delta'])\cap
Y_\Delta$, and
$\emptyset$ otherwise.
\end{defn}

The next theorem from \cite{comb_HHS} provides the criteria we will use to prove that $\bar E$ has an HHS structure.

\begin{theorem}{\cite[Theorem 1.18]{comb_HHS}}\label{thm:hhs_links}
Let $(X,W)$ be a combinatorial HHS.
Let $\mathfrak S$ be as in Definition~\ref{defn:simplex_equivalence}, define nesting and orthogonality
relations on $\mathfrak S$ as in Definitions~\ref{defn:nest} and~\ref{defn:orth}, let the associated hyperbolic spaces be as in
Definition~\ref{defn:combinatorial_HHS}, and define projections as in Definition~\ref{defn:projections}.
Then $(W,\mathfrak S)$ is a hierarchically hyperbolic space, and the HHS constants only depend on $n$, $\delta$ as in Definition
\ref{defn:combinatorial_HHS}.
\end{theorem}

\subsection{Combinatorial HHS structure}\label{subsec:our_comb_HHS}
 We now define a flag simplicial complex $\CX$. The vertex set is $\CX^{(0)}= \vtx \sqcup \kvtx$, where
  \[ \kvtx = \bigsqcup_{v \in \vtx} \mathcal K^v. \]
Given a vertex $s \in \kvtx$, let $v(s) \in \vtx$ be the unique vertex with $s \in \CK^v$.  We also write $\alpha(s)= \alpha(v(s))$.

There are 3 types of edges (see Figure~\ref{fig:blowup}):
 \begin{enumerate}
 \item $v,w \in \vtx$ are connected by an edge if and only if $d_{\tree}(v,w) = 1$.
 \item $s,t \in \kvtx$ are connected by an edge if and only if $d_{\tree}(v(s),v(t))=1$.
\item  $s \in \kvtx$ and $w \in \vtx$ are connected by an edge if and only if $d_{\tree}(v(s),w) \leq 1$.
 \end{enumerate}
We declare $\CX$ to be the flag simplicial complex with the 1-skeleton defined above.

\begin{figure}[h]
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\draw[thick,->>] (3,1.5) .. controls (5,2)  .. (7,1.5);

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\draw[ultra thick] (9,0) -- (11,-2);
\draw[ultra thick] (9,0) -- (7,-2);

\node[draw,fill=lime,circle] at (9,3) {\small $v_1$};
\node[draw,fill=yellow,circle] at (9,0) {\small $v_2$};
\node[draw,fill=pink,circle] at (7,-2) {\small $v_3$};
\node[draw,fill=cyan,circle] at (11,-2) {\small $v_4$};

\node at (-1.25,0) {$v(s_i) = $};
\node at (-1.25,3) {$v(t_i)=$};

\node at (5,2.4) {$Z$};

\end{tikzpicture}
\caption{The simplicial map $Z$ restricted to a part of $\CX$ (on the left) to a part of the union of trees $T_\alpha$ (on the right).}\label{fig:blowup}

\end{figure}

The map $\kvtx \to \vtx$ given by $s \mapsto v(s)$ and the identity $\vtx \to \vtx$ extends to a surjective simplicial map
 \[ Z \colon \CX \to \bigsqcup_{\alpha \in \CP} T_\alpha. \]
We note that we may view the union $\bigsqcup T_\alpha$ on the right as a subgraph of $\CX^{(1)}$, making $Z$ a retraction.

For any vertex $v$ in any tree $T_\alpha$, $Z^{-1}(v)$ is the join of $\{v\}$ and the set $\mathcal K^v$:
\begin{equation} \label{E: Z preimage of vertex} Z^{-1}(v) = \{v\} \star \mathcal K^v. \end{equation}
For any pair of adjacent vertices $v,w \in T_\alpha$ (so $d_{\tree}(v,w) = 1$), the preimage of the edge $[v,w] \subset T_\alpha$ is also a join:
\begin{equation} \label{E: Z preimage of edge} Z^{-1}([v,w]) = Z^{-1}(v) \star Z^{-1}(w) = (\{v\} \star \mathcal K^v) \star (\{w\} \star \mathcal K^w).\end{equation}

 \begin{lemma}\label{lem:max_simplex}
  The maximal simplices of $\CX$ are exactly the $3$--simplices with vertex set $\{s,v(s),t,v(t)\}$ where $s,t \in \kvtx$ and $d_{\tree}(v(s),v(t))=1$. In this case, we say that $(s,t)$ defines a maximal simplex, denoted $\sigma(s,t)$.
 \end{lemma}

 \begin{proof}
 Because the map $Z$ is simplicial, any simplex of $\CX$ is contained in $Z^{-1}(v)$ or $Z^{-1}([v,w])$ for some vertex $v$ in some $T_\alpha$ or some edge $[v,w]$ in some $T_\alpha$.  The lemma thus follows from \eqref{E: Z preimage of vertex} and \eqref{E: Z preimage of edge}.
 \iffalse
  Consider a simplex $\Delta$ with vertices $\{s,v(s),t,v(t)\}$ as described in the statement. If $\Delta$ was not maximal, then there would be a vertex $w$ of type 1 connected to all vertices of $\Delta$, by Remark \ref{rem:same_link}. But since $T_{\alpha(v)}$ is a tree and $d_{\tree}(v,w)=1$, there is no vertex $w$ in $T_{\alpha(v)}$ with $d_{\tree}(v(s),w)=d_{\tree}(v(t),w)=1$. This proves that $\Delta$ is maximal.

  Conversely, consider a maximal simplex $\Delta$. By Remark \ref{rem:same_link}, if the vertex set of $\Delta$ contains a vertex $s$ of type 2, then it also contains $v(s)$. Therefore, the vertex set of $\Delta$ is of the form $\{s_1,v(s_1),\dots,s_n,v(s_n)\}$. Since every vertex of type 1 has a neighbor, we have $n\geq 2$. Also, notice that $\{s_1,v(s_1),s_2,v(s_2)\}$ is the vertex set of a simplex as in the statement, and we just argued that such simplices are maximal. Therefore, $n=2$ and we are done.
  \fi
 \end{proof}

Given a vertex $s \in \kvtx$, recall from \S\ref{S:Kapovich-Leeb distances} that $M(s)=(\lambda^{v(s)})^{-1}(s) \subset \bTheta^{v(s)}$.  Given a pair of vertices $(s,t)$ in $\kvtx$ that define a maximal simplex $\sigma(s,t)$, we will write
$M(s,t) = M(s) \cap M(t)$.
\begin{lemma}\label{lem:dense}
There exists $R>0$ with the following properties.
\begin{enumerate}
 \item \label{item:diam intersect} For any pair of adjacent vertices $s,t\in \kvtx$ (i.e., defining a maximal simplex $\sigma(s,t)$), $M(s,t)$ is a non-empty subset of diameter at most $R$.
 \item \label{item:dense bTheta} Given $v\in \vtx$, we have
 $$\bTheta^v=\bigcup_{v(s)=v,d_\tree(v(s),v(t))=1} M(s,t).$$
 \item \label{item:dense in KL} Fixing $s$, we have
 $$M(s)=\bigcup_{d_\tree(v(s),v(t))=1}M(s,t).$$
  \item \label{item:dense bar E} The collection of all $M(s,t)$ is $R$--dense in $\bar{E}$.
\end{enumerate}
\end{lemma}

\begin{proof}
 Item \eqref{item:diam intersect} follows from Proposition \ref{prop:KL}\eqref{item:KL_qi}. More precisely, the fact that $M(s,t)$ is non-empty follows from surjectivity of $\lambda^{v(s)}\times\lambda^{v(t)}$, while boundedness follows from the fact that said map is a quasi-isometry.

 In order to show item \eqref{item:dense bTheta}, notice that
 \[ \bTheta^v=\bigcup_{d_{\tree}(v,w)=1} \bTheta^v\cap\bTheta^w.\]
That is, every point of $\bTheta^v$ is also in $\bTheta^w$ for some $w$ adjacent to $v$. In view of this, we conclude by noticing that if $x\in \bTheta^v\cap\bTheta^w$, then $x\in p^{-1}(\lambda^v(x), \lambda^w(x))$. Item \eqref{item:dense in KL} follows similarly.

 Finally, item \eqref{item:dense bar E} follows from item \eqref{item:dense bTheta} and the fact that the collection of all $\bTheta^v$ is coarsely dense in $\bar E$.
\end{proof}

Next we define a graph $\CW$ whose vertex set is the set of maximal simplices of $\CX$. We would like to just connect maximal simplices when the corresponding subsets $M(s,t)$ are close in $\bar E$ (first bullet below); however, in order to arrange item \eqref{item:C_0=C} of the definition of combinatorial HHS (and only for that reason) we need different closeness constants for different situations.  We fix $R$ as in Lemma \ref{lem:dense}, and moreover we require $R>K_1^2+K_1$, for $K_1$ as in Proposition \ref{prop:KL}.

Given maximal simplices $\sigma(s_1,t_1)$ and $\sigma(s_2,t_2)$, we declare them to be connected by an edge in $\CW$ if one of the following holds:
\begin{equation}\label{Eq:Def W-edges}
\begin{array}{l} \bullet \, \,  \bar d(M(s_1,t_1),M(s_2,t_2))\leq 10 R \hspace{5cm} \\
 \bullet \,\, s_1=s_2 \mbox{ and } \bar d(M(t_1),M(t_2))\leq 10 R \end{array}
 \end{equation}
  Here the the $\bar d$--distances are the {\em infimal} distances between the sets in $\bar E$ (as opposed to the diameter of the union).
  Note that since $M(s,t)=M(t,s)$, the second case also implicitly describe a ``symmetric case'' with $s_i$ and $t_i$ interchanged.

The following is immediate from Lemma \ref{lem:retract_on_the_middle}, setting $R'=\max\{10R,N_{\mathcal K}(10R)\}$.

\begin{lemma}\label{lem:connect_not_far}
 There exists $R'\geq 10R$ so then the following holds. If $s,t_1,t_2\in \kvtx$ are vertices with $s$ connected to both $t_i$ in $\CX$ and $\bar d(M(t_1),M(t_2))\leq 10 R$, then $\bar d(M(s,t_1),M(s,t_2))\leq R'$. In particular, whenever $\sigma(s_1,t_1)$ and $\sigma(s_2,t_2)$ are connected in $\CW$, we have $\bar d(M(s_1,t_1),M(s_2,t_2))\leq R'$.
\end{lemma}

\begin{lemma}\label{lem:W=barE}
 $\CW$ is quasi-isometric to $\bar E$, by mapping each $\sigma(s,t)$ to (any point in) $M(s,t)$.
\end{lemma}

\begin{proof}
 In view of Lemma \ref{lem:connect_not_far}, the lemma follows by combining Lemma \ref{lem:dense}(\ref{item:dense bar E}) and Proposition \ref{P:graph approximation} (applied to any choice of a point in each $M(s,t)$).
 \end{proof}

The goal for the remainder of this section is to prove the following.

\begin{theorem}\label{thm:E_is_chhs}
The pair  $(\CX,\CW)$ is a combinatorial HHS.
\end{theorem}

\subsection{Simplices, links, and saturations}
\label{subsec:links_list}
Before giving the proof of Theorem~\ref{thm:E_is_chhs}, we begin by describing explicitly the kinds of simplices of $\CX$ that there are, explain what their links and saturations are, and observe some useful properties.

\begin{lemma} [Empty simplex] \label{L: empty simplex case}
For the empty simplex, $\mathcal C(\emptyset)=\CX^{+\CW}$ is quasi-isometric to $\hat{E}$.
\end{lemma}
\begin{proof} We define a map $Z' \colon \CX^{+\CW} \to \hat E$ that extends the (restricted) simplicial map $Z \colon \CX^{(1)} \to \bigsqcup T_\alpha$ already constructed above.   To do that, we must extend over each edge $e=[x,y]$ of $\CX^{+\CW}$ coming from the edge of $\CW$ connecting $\sigma(s_1,t_1)$ and $\sigma(s_2,t_2)$.  Since $Z(x),Z(y) \in \vtx$, and $\bar d(M(s_1,t_1),M(s_2,t_2))\leq R'$ (for $R'$ as in Lemma \ref{lem:connect_not_far}), we see that $d_{\hat E}(v,w)\leq R'$.  We can then define $Z'$ on $e$ to be a constant speed parameterization of a uniformly bounded length path from $Z(x)$ to $Z(y)$.  It follows that $Z'$ is Lipschitz.

The union of the trees $\bigsqcup T_\alpha$ is $R_0$--dense (Lemma~\ref{L: collapsed vertex set dense}), so it suffices to find a one-sided inverse to $Z'$, from $\bigsqcup T_\alpha$ to $\CX^{+\CW}$ and show that with respect to the subspace metric from $\hat E$, it is coarsely Lipschitz.  As already noted, $Z$ restricts to a retraction of $\CX^{(1)}$ onto $\bigsqcup T_\alpha \subset \CX^{(1)} \subset \CX^{+\CW}$, which is thus the required one-sided inverse.  All that remains is to show that it is coarsely Lipschitz.
In view of Lemma~\ref{L: combinatorial path}, any $v\in T_\alpha,w\in T_\beta$ are connected by a combinatorial path (see Definition \ref{D:combinatorial path}) of length comparable to $\hat{d}(v,w)$, and consisting of a number of horizontal jumps, each of uniformly bounded length, and with total length bounded in terms of $\hat{d}(v,w)$.
Therefore, we can reduce to the case that $v,w$ are joined by a single horizontal jump of bounded length. Such a horizontal jump can also be regarded as a path in $\bar E$ connecting $\bTheta^v$ to $\bTheta^w$. Hence, in view of Lemma \ref{lem:dense}\eqref{item:dense bTheta}, there are $M(s_1,t_1)\subseteq \bTheta^v$ and $M(s_2,t_2)\subseteq \bTheta^w$ within uniformly bounded distance of each other in $\bar E$. Lemma \ref{lem:W=barE} implies that there exists a path in $\CW$ of uniformly bounded length from $\sigma(s_1,t_1)$ to $\sigma(s_2,t_2)$, which can be easily turned into a path of uniformly bounded length from $v$ to $w$ in $\CX^{+\CW}$, as required.
\end{proof}

There is an important type of $1$--dimensional simplex, which we call this a {\em $\bXi$--type simplex}, due to the following lemma.  See Figure~\ref{F:link-sat-Xi}.  Given $w \in \vtx$, set
\begin{equation} \label{E:Xi link} \link_{\bXi}(w) = \bigcup_{d_{\tree}(u,w) = 1} Z^{-1}(u) = \bigcup_{d_{\tree}(u,w) = 1} \{u\} \star \CK^u.
\end{equation}

\begin{lemma} [$\bXi$--type simplex]  \label{L: Xi simplex}
Let $\Delta$ be a $1$--simplex of $\CX$ with vertices $s,v(s)$, for $s \in \kvtx$. Then
\[ \link(\Delta) = \link_{\bXi}(v(s)) \quad \mbox{ and } \quad \Sat(\Delta)=\{v(s)\}\cup \CK^{v(s)}.\]
Moreover, $\mathcal C(\Delta)$ is quasi-isometric to $\bXi^{v(s)}$, via a quasi-isometry which is the identity on $\vtx \cap \mathcal C(\Delta)$ and maps $t$ to $v(t)$ for $t \in \kvtx \cap \C(\Delta)$.
\end{lemma}

\begin{figure}[h]
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\node at (1,-4) {$\sat(\Delta)$};

\iffalse
\begin{scope}[shift={(6,0)}]

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\node at (1,-4) {$\link(\Delta) = \link_{\bXi}(v)$};

\end{scope}

\end{tikzpicture}
\caption{$\bXi$--type simplex: Part of the link and saturation of a $1$--simplex $\Delta$ with vertices $s,v=v(s)$.} \label{F:link-sat-Xi}
\end{figure}
\begin{proof} It is clear from the definitions that the link of $\Delta$ is as described. Also, any simplex with vertex set of the form $\{v(s),t\}$ for some $t \in \kvtx^{v(s)}$ has the same link as $\Delta$. Therefore, to prove that the saturation is as described we are left to show if a simplex $\Delta'$ has the same link as $\Delta$, then its vertex set is contained in the set we described. If $w \in \vtx$ is a vertex of $\Delta'$, then $d_{\tree}(v,w)=1$ for all neighbors $v$ of $v(s)$ in $T_{\alpha(v)}$. This implies $w=v(s)$. Similarly, if $t \in \kvtx$ is a vertex of $\Delta'$, then $v(t)=v(s)$, and we are done.

Let us show that the map given in the statement is coarsely Lipschitz. To do so, it suffices to consider $w \neq w'$ with $d_{\tree}(w, v(s))=d_{\tree}(w', v(s))=1$ and connected by an edge in $\CX^{+\CW}$, and show that they are connected by a bounded-length path in $\bXi^v$. We argue below that $\bar d(\bTheta^w\cap \bTheta^{v(s)},\bTheta^{w'}\cap \bTheta^{v(s)})\leq R'$. Once we do that, the existence of the required bounded-length path follows directly from Lemma \ref{lem:parameter_for_Xi}.

 Let us now prove the desired inequality. Notice that $w,w'$ cannot be connected by an edge of $\CX$ since they are both distance 1 from $v(s)$ in the tree $T_{\alpha(v)}$. Hence, $w$ and $w'$ are contained respectively in maximal simplices $\sigma(t,u)$ and $\sigma(t',u')$ connected by an edge in $\mathcal W$. Say, up to swapping $t$ with $u$ and/or $t'$ with $u'$, that $v(t)=w$ and $v(t')=w'$. In either case of the definition of the edges of $\CW$ we have $\bar d(M(t),M(t'))\leq 10 R$. Since $M(t,s)\subseteq \bTheta^w\cap \bTheta^{v(s)}$ and $M(t',s)\subseteq \bTheta^{w'}\cap \bTheta^{v(s)}$, using Lemma \ref{lem:connect_not_far} we get
  $$\bar d(\bTheta^w\cap \bTheta^{v(s)},\bTheta^{w'}\cap \bTheta^{v(s)})\leq \bar d(M(t,s),M(t',s))\leq R',$$
  as we wanted.

Conversely, if $w,w'$ as above are joined by an edge in $\bXi^{v(s)}$ we will now show that they are also connected by an edge in $\mathcal C(\Delta)$. By definition of $\bXi^{v(s)}$, we have
$\bTheta^w\cap\bTheta^{w'}\neq\emptyset$.
By Lemma~\ref{lem:dense}\eqref{item:dense bTheta}, There exists $t,u,t',u'$ with $v(t) = w$ and $v(t') = w'$, so that $M(t,u) \cap M(t',u') \neq \emptyset$.
In particular, $\bar d(M(t,u),M(t',u'))\leq 10 R$.
This says that $\sigma(t,u)$ and $\sigma(t',u')$ are connected in $\CW$, and hence that $w$ and $w'$ are connected in $\CX^{+\CW}$, as required.
\end{proof}

There is also an important type of $2$--dimensional simplex, which we call this a {\em $\mathcal K$--type simplex}, due to the next lemma. See Figure~\ref{F:link-sat-KL}.

\begin{lemma} [$\mathcal K$--type simplex]  \label{L: KL simplex}
Let $\Delta$ be a $2$--simplex of $\CX$ with vertices $s,v(s),w$, for $s \in \kvtx$ and $w \in \vtx$ with $d_{\tree}(w,v(s))=1$. Then
\[ \link(\Delta) = \CK^w \quad \mbox{ and } \quad \Sat(\Delta) = \{w\} \cup \link_{\bXi}(w)^{(0)}.\]
Moreover, $\mathcal C(\Delta)$ is quasi-isometric to $\mathcal K^w$, the quasi-isometry being the identity at the level of vertices.
\end{lemma}
\begin{figure}[h]
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\caption{$\mathcal K$--type simplex: Part of the link and saturation of a $2$--simplex $\Delta$ with vertices $s,v=v(s),w$.}\label{F:link-sat-KL}
\end{figure}
\begin{proof}
It is clear from the definitions that the link of $\Delta$ is as described. Also, any simplex with vertex set of the form $\{t,v(t),w\}$ for some $t \in \kvtx$ with $d_{\tree}(v(t),w)=1$ has the same link as $\Delta$. Therefore, to prove that the saturation is as described we are left to show if a simplex $\Delta'$ has the same link as $\Delta$, then its vertex set is contained in the set we described. Given any vertex $u \in \vtx$ of $\Delta'$, it has to be connected to all $t \in \kvtx$ with $v(t)=w$, implying that either $u=w$ or $d_{\tree}(u,w)=1$, as required for vertices in $\vtx$. Similarly, any vertex $u \in \kvtx$ of $\Delta'$ has to be connected to all $t \in \kvtx$ with $v(t)=w$, implying $d_{\tree}(v(u),w)=1$, and we are done.

 To prove that $\mathcal C(\Delta)$ is naturally quasi-isometric to $\mathcal K^w$, it suffices to show that if $s,t\in\mathcal C(\Delta)$ are connected by an edge in $\CX^{+\CW}$, then they are uniformly close in $\mathcal K^w$ and that, vice versa, if $d_{\mathcal K^w}(s,t)=1$, then $s,t$ are connected by an edge in $\mathcal C(\Delta)$.

 First, if $s,t\in\mathcal C(\Delta)$ are connected by an edge in $\CX^{+\CW}$, then there exist $s',t'$ so that $\bar d(M(s,t'),M(s',t))\leq R'$ (see Lemma \ref{lem:connect_not_far}). In particular, $\bar d(M(s),M(t))\leq R'$, which in turn gives a uniform bound on the distance in the path metric of $\bTheta^w$ between $M(s)$ and $M(t)$ because the metrics $\bar{d}$ and the path metric on $\theta^w$ are coarsely equivalent. By Proposition \ref{prop:KL}(\ref{item:lambda_lipschitz}) we must also have a uniform bound on $d_{\mathcal K^w}(s,t)$.

 Now suppose that $d_{\mathcal K^w}(s,t)=1$.  We can then deduce from Proposition \ref{prop:KL}(\ref{item:KL_qi}) that implies that $\bar d(M(s',s),M(s',t))\leq K_1^2+K_1\leq 10R$ for any fixed $s'\in \mathcal K^{v(s)}$ (the proposition yields the analogous upper bound in the path metric of $\bTheta^w$, which is a stronger statement). Therefore $s,t$ are connected by an edge in $\CX^{+\CW}$, whence in $\mathcal C(\Delta)$, as required.
\end{proof}

The remaining simplices are not particularly interesting as their links are joins (or points), and hence have diameter at most $3$, but we will still need to verify properties for them.  We describe each type and its link and saturation in the next lemma. We also add to the lemma another property that we need later. Recall the definition of $\link_{\bXi}(w)$ in \eqref{E:Xi link}.

\begin{lemma} \label{L:remaining simplices} The following is a list of all types of non-empty, nonmaximal, simplices $\Delta$ of $\CX$ that are not of $\bXi$--type or $\mathcal K$--type, together with their links.  In each case, the link is a nontrivial join (or a point), and $\mathcal C(\Delta)$ has diameter at most $3$.

In the table below the simplices $\Delta$ have vertices $u,w \in \vtx$  with $d_{\tree}(u,w) = 1$ and $s,t \in \kvtx$ with $d_{\tree}(v(s),v(t)) = 1$ and $d_{\tree}(v(s),u) =1$.\\
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$\Delta$ & $\link(\Delta)$ \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{u\}\phantom{\frac{\int_1^2}{\int_1^2}}$ & \quad $\CK^u \star \link_\bXi(u)$ \quad \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{s\}\phantom{\frac{\int_1^2}{\int_1^2}}$ & \quad $\{v(s)\} \star\link_\bXi(v(s))$ \quad \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{u,w\}\phantom{\frac{\int_1^2}{\int_1^2}}$ & $\CK^u \star \CK^w$ \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{s,t\}\phantom{\frac{\int_1^2}{\int_1^2}}$ & $\{ v(s),v(t) \}$ \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{s,u\}\phantom{\frac{\int_1^2}{\int_1^2}}$  & $\{v(s)\} \star \CK^u$ \\

\hline
$\phantom{\frac{\int_1^2}{\int_1^2}}\{s,t,v(s)\}\phantom{\frac{\int_1^2}{\int_1^2}}$ & $\{v(t)\}$ \\

\hline
\end{tabular}
\end{center}

\medskip

\end{lemma}
\begin{proof} This is straightforward given the definition of $\CX$ and we leave its verification to the reader.  Referring to Figure~\ref{fig:blowup}, and comparing with Figures~\ref{F:link-sat-Xi} and \ref{F:link-sat-KL}, may be helpful.
\end{proof}
\iffalse
\begin{corollary} \label{C:link joins} The following holds for all simplices $\Delta$ of $\CX$.
\begin{enumerate}
\item \label{item:big link of a join} If $\Gamma$ is a simplex of $\bXi$--or $\mathcal K$--type and $\link(\Gamma)\subseteq \link(\Delta)$, then
\[ \link(\Gamma)=\link(\Delta*\Pi)\]
for some (possibly empty) simplex $\Pi$ contained in $\link(\Delta)$.
\item \label{item:joined big links} If $\Gamma_1,\Gamma_2$ are simplices of $\bXi$--or $\mathcal K$--type, $\link(\Gamma_i)\subseteq \link(\Delta)$, and $\link(\Gamma_1)$ forms a join with $\link(\Gamma_2)$, then
\[ \link(\Gamma_1)*\link(\Gamma_2)=\link(\Delta*\Pi)\]
for some (possibly empty) simplex $\Pi$ contained in $\link(\Delta)$.
\end{enumerate}
\end{corollary}
\begin{proof} Both parts of this corollary follow from Lemmas~\ref{L: Xi simplex}, \ref{L: KL simplex}, and \ref{L:remaining simplices} and case-by-case considerations.
\end{proof}
\fi
The next lemma collects a few additional properties we will need.  By the {\em type} of a simplex in $\CX$, we mean its orbit by the action of the simplicial automorphism group of $\CX$.  There are $9$ types of nonempty simplices: maximal, $\bXi$--type, $\CK$--type, and the six types listed in Lemma~\ref{L:remaining simplices}.

\begin{lemma}  \label{L: nice join properties }
The following hold in $\CX$.
\begin{enumerate}[$(a)$]
\item The link of a simplex with a given type cannot be strictly contained in the link of a simplex with the same type.
   \item For every non-maximal simplices $\Delta$ and $\Delta'$ so that there is a simplex $\Gamma$ with $\link(\Gamma)\subseteq \link(\Delta')\cap \link(\Delta)$ and $diam(\mathcal C (\Gamma))>3$, there exists a simplex $\Pi$ in the link of $\Delta'$ with $\link(\Delta'*\Pi)\subseteq \link(\Delta)$ so that for any $\Gamma$ as above we have $\link(\Gamma)\subseteq \link(\Delta'*\Pi)$.
 \end{enumerate}
\end{lemma}
\begin{proof} Part $(a)$ follows directly from the descriptions of the simplices given in Lemmas~\ref{L: Xi simplex}, \ref{L: KL simplex}, and \ref{L:remaining simplices}, and we leave this to the reader.

Before we prove $(b)$, we suppose $\link(\Delta') \cap \link(\Delta) \neq \emptyset$, and make a few observations.  First, $\Delta'$ and $\Delta$ must project by $Z$ to the same tree: $Z(\Delta'),Z(\Delta) \subset T_\alpha$ for some $\alpha \in \mathcal P$.  Next, note that $Z(\link(\Delta') \cap \link(\Delta))$ is contained in the intersection of the stars in $T_\alpha$ of $Z(\Delta')$ and $Z(\Delta)$.  Moreover, (as in any tree) the intersection of these two stars is contained in a single edge, or else $Z(\Delta') = Z(\Delta) = \{w\} \in T_\alpha^{(0)} \subset \vtx$ is a single point.  In this latter case, by \eqref{E: Z preimage of vertex}, we have
\[ \Delta,\Delta' \subset \{w\} \star \CK^w.\]

Next, note that for any $\CK$--type simplex $\Gamma = \{s,v(s),w\}$,  $\link(\Gamma) = \CK^w$ by Lemma~\ref{L: KL simplex}, and if $\link(\Gamma) \subset \link(\Delta') \cap \link(\Delta)$, then $w$ is in the intersection of the stars of $Z(\Delta')$ and $Z(\Delta)$.  For a $\bXi$--type simplex $\Gamma = \{s,v(s)\}$, $\link(\Gamma) = \link_{\bXi}(v(s))$ by Lemma~\ref{L: Xi simplex}, and together with Lemma~\ref{L:remaining simplices} and the previous paragraph, we see that $\link(\Gamma) \subset \link(\Delta') \cap \link(\Delta)$ if and only if $Z(\Delta') = Z(\Delta) = \{v(s)\}$ in $T_\alpha$.

With these observations in hand, we proceed to the proof of $(b)$, which divides into two cases.

\smallskip

\noindent {\bf Case 1.} There is a $\bXi$--type simplex $\Gamma = \{s,v(s)\}$ with $\link(\Gamma) \subset \link(\Delta') \cap \link(\Delta)$.

\smallskip

In this situation, $Z(\Delta') = Z(\Delta) = \{v(s)\}$, and thus $\Delta,\Delta' \subset \{v(s)\} \star \CK^{v(s)}$.
From Lemmas~\ref{L: Xi simplex} and \ref{L:remaining simplices}, we see that $\link(\Delta') \cap \link(\Delta)$ must be equal to one of $\link(\Gamma)$, $\link(\{s\})$, or $\link(\{v(s)\})$.  Inspection of these links shows that $\link(\Gamma)$ is the only link of a $\bXi$--type simplex contained in it.  First suppose that $\link(\Delta') \cap \link(\Delta)$ has the form $\link(\Gamma)$ or $\link(\{s\})$.  In this situation, we easily find $\Pi \subset \link(\Delta')$ so that $\link(\Gamma) = \link(\Delta' \star \Pi)$.  Furthermore, for any $\CK$--type simplex link $\CK^w$ in the intersection, we must have $\CK^w \subset \link(\Gamma) = \link(\Delta' \star \Pi)$.  Therefore, the link of any $\bXi$--or $\CK$--type simplex contained in $\link(\Delta') \cap \link(\Delta)$ must be contained in $\link(\Gamma) = \link(\Delta' \star \Pi)$, as required.  Now suppose instead that $\link(\Delta') \cap \link(\Delta) = \link(\{v(s)\})$.  By Lemma~\ref{L:remaining simplices}, we see that $\Delta = \Delta' = \{v(s)\}$.  In this case, setting $\Pi = \emptyset$ trivially completes the proof since then $\link(\Delta') \cap \link(\Delta) = \link(\Delta') = \link(\Delta'\star \Pi)$.

\smallskip

\noindent {\bf Case 2.} No link of a $\bXi$--type simplex is contained in $\link(\Delta') \cap \link(\Delta)$.

\smallskip

From the observations above, $Z(\Delta)$ and $Z(\Delta')$ do not consist of the same single point, and hence the stars of $Z(\Delta)$ and $Z(\Delta')$ intersect in either a point or an edge in $T_\alpha$.  Since $Z(\link(\Delta') \cap \link(\Delta))$ is contained in the intersection of these stars, there are at most two $\CK$--type simplices whose links are contained in $\link(\Delta') \cap \link(\Delta)$.  If there are two $\CK$--type simplices $\Gamma,\Gamma'$ with $\CK^w = \link(\Gamma)$ and $\CK^u = \link(\Gamma')$ contained in $\link(\Delta') \cap \link(\Delta)$, then observe that
\[ \CK^u,\CK^w \subset \link(\{u,w\}) = \CK^u \star \CK^w \subset \link(\Delta') \cap \link(\Delta).\]
By inspection of the possible links in Lemmas~\ref{L: Xi simplex}, \ref{L: KL simplex}, and \ref{L:remaining simplices}, it must be that $\Delta'$ is either $\{u\}$, $\{w\}$, or $\{u,w\}$, and so setting $\Pi$ to be $\{w\}$, $\{u\}$, or $\emptyset$, respectively, we are done.  On the other hand, if there is exactly one $\CK$--type simplex $\Gamma$ with $\CK^w = \link(\Gamma) \subset \link(\Delta') \cap \link(\Delta)$, then again inspecting all possible situations, we can find $\Pi \subset \link(\Delta')$ with $\CK^w =\link(\Gamma) = \link(\Delta' \star \Pi)$, and again we are done with this case.  This completes the proof.
\end{proof}

\iffalse
Suppose first that $\link(\Delta')\cap \link(\Delta)$ does not contain a link of a simplex of $\bXi$--type. Then we claim that $\link(\Delta')\cap \link(\Delta)$ either contains exactly one link of a simplex of $\mathcal K$--type, or it contains two of them that form a join. In particular, the required $\Pi$ is provided by Corollary~\ref{C:link joins}, parts \eqref{item:big link of a join} and \eqref{item:joined big links}, respectively.  To verify the claim, recall that the link of a $\CK$--type simplex has the form $\CK^u$, for $u \in \vtx$.  Therefore, suppose the claim does not hold, and hence there must be $u,w \in \vtx$ with $d_{\tree}(u,w) \geq 2$ so that $\CK^u,\CK^w \subset \link(\Delta') \cap \link(\Delta)$.
Now consider the projections $Z(\Delta'),Z(\Delta) \subset T_\alpha$ of $\Delta'$ and $\Delta$ to the appropriate tree $T_\alpha$, and notice that the intersection of the stars in $T_\alpha$ of $Z(\Delta'),Z(\Delta)$ contains $Z(\link(\Delta')\cap \link(\Delta))$.  But then $u,w \in Z(\link(\Delta') \cap \link(\Delta))$, which are non-adjacent vertices of $T_\alpha$, which implies $Z(\Delta')= Z(\Delta)$ is a single point. In this case, consulting Lemma~\ref{L:remaining simplices} we see that $\link(\Delta')\cap \link(\Delta)$ contains a link of a simplex of $\bXi$--type, which is a contradiction.

 Suppose now that $\link(\Delta')\cap \link(\Delta)$ contains a link $\link(\Gamma_0)$ of a simplex of $\bXi$--type.  A similar argument as above shows that $\Delta',\Delta$ project to a single point in the appropriate $T_\alpha$. In this case it is easily seen that either there is no $\link(\Gamma)$ contained in $\link(\Delta')\cap \link(\Delta)$ but not contained in $\link(\Gamma_0)$, or there is exactly one and it forms a join with $\link(\Gamma_0)$. In either case, the required $\Pi$ is provided by the moreover part of Lemma \ref{L:remaining simplices}.
\fi

\begin{lemma}\label{lem:retractions}
 There exists $L\geq 1$ so that for every non-maximal simplex, there is an $(L,L)$--coarsely Lipschitz retraction $r_\Delta\colon Y_\Delta\to \mathbb{P}({\mathcal C(\Delta)})$. In particular, $\mathcal C(\Delta)$ is uniformly quasi-isometrically embedded in $Y_\Delta$.
\end{lemma}

\begin{proof} By Lemma~\ref{L:remaining simplices}, we only have to consider simplices of $\bXi$--and $\mathcal K$--type.

Consider $\Delta = \{s,v\}$ with $v = v(s)$ of $\bXi$--type first.  Recall from Lemma~\ref{L: Xi simplex} that $\bXi^v$ naturally includes into $\C(\Delta)$ by a quasi-isometry.  Here we will use make use of the map $\xi^v$, whose relevant properties for our current purpose are stated in Proposition \ref{prop:R-projections}. For a vertex $u \in \vtx \smallsetminus \Sat(\Delta)$ (so, $u \neq v$) we define $r_\Delta(u)=\xi^v(\bTheta^{u})$.  For $t \in \kvtx \smallsetminus \Sat(\Delta)$ we define $r_\Delta(t)=\xi^v(\bTheta^{v(t)})$. Notice that the sets $r_\Delta(u)$ are uniformly bounded by Proposition \ref{prop:R-projections} (and Lemma~\ref{L: Xi simplex}).  Also, $r_\Delta$ is coarsely the identity on the vertices of $\link(\Delta)$ in $\vtx$ by Equation~\eqref{E:an obvious but useful fact} and Proposition \ref{prop:R-projections}(2b).  To check that $r_\Delta$ is coarsely Lipschitz it suffices to consider $\CX^{+\CW}$--adjacent vertices of $\vtx$. Notice that vertices $w,w' \in \vtx$ that are adjacent in $\CX^{+\CW}$ have corresponding $\bTheta^w,\bTheta^{w'}$ within $10R$ of each other in $\bar E$. Indeed, $\bTheta^v$ and $\bTheta^w$ actually intersect if $w,w'$ are adjacent in $\CX$, and they contain subsets $M(\cdot)$ within $10R$ of each other if $w,w'$ are contained in $\CW$--adjacent maximal simplices (this is true regardless of which case of the definition \eqref{Eq:Def W-edges} for the edges of $\CW$ applies). The fact that $r_\Delta$ is coarsely Lipschitz now follows from, Proposition~\ref{prop:R-projections}(1), which says that $\xi^v$ is coarsely Lipschitz on $\bar E$.

Next, consider $\Delta = \{s,v(s),w\}$ of $\mathcal K$--type.  For a vertex $u \in \vtx \smallsetminus \Sat(\Delta)$ (so, $d_{\tree}(u,w) \geq 2$), define $r_\Delta(u)=\Lambda^w(\bTheta^u)$.  For a vertex $t \in \kvtx \smallsetminus \Sat(\Delta)$, define $r_\Delta(t)=\Lambda^w(M(t))$.  Notice that, by definition, if $t \in \CK^w$, then $r_\Delta(t)=t$.  Also, since $d_{\tree}(u,w) \geq 2$, Proposition~\ref{prop:R-projections}(2a) ensures that the diameter of $\Lambda^w(\bTheta^u)$ is bounded. Since $M(t)\subseteq \bTheta^{v(t)}$, we see that all the sets in the image of $r_\Delta$ are bounded, and also we see that in order to prove that $r_\Delta$ is Lipschitz it suffices to consider vertices of $\kvtx$.  But vertices $s,t \in \kvtx$ that are adjacent in $\CX^{+\CW}$ have corresponding $M(s),M(t)$ within $10R$ of each other in $\bar E$, so the conclusion follows from Proposition~\ref{prop:R-projections}(1), which states that $\Lambda^w$ is coarsely Lipschitz on $\bar E$.
\end{proof}

\subsection{Final proof}\label{subsec:final_proof} We now have all the tools necessary for the:

\begin{proof}[Proof of Theorem \ref{thm:E_is_chhs}]  We must verify each of the conditions from Definition~\ref{defn:combinatorial_HHS}.

Item \eqref{item:chhs_flag} (bound on length of $\nest$--chains) follows from Lemma~\ref{L: nice join properties }(a), which implies that any chain $\link(\Delta_1)\subsetneq \dots$ has length bounded by the number of possible types, which is 9.

 Let us now discuss item \eqref{item:chhs_delta} of the definition. The descriptions of the $\mathcal C(\Delta)$ from Lemmas~\ref{L: empty simplex case}, \ref{L: Xi simplex}, \ref{L: KL simplex}, \ref{L:remaining simplices}  yields that all $\mathcal C(\Delta)$ are hyperbolic, since each of them is either bounded or uniformly quasi-isometric to one of $\hat{E}$ (which is hyperbolic by Theorem \ref{T:hyperbolicity of hat E}), some $\bXi^v$ (which is hyperbolic by Lemma \ref{lem:Xi_hyp}), or $\mathbb R$. Moreover, any $\mathcal C(\Delta)$ is (uniformly) quasi-isometrically embedded in $Y_\Delta$ by Lemma \ref{lem:retractions}.

Item \eqref{item:chhs_join} of the definition (common nesting) is precisely Lemma \ref{L: nice join properties }(b).

Finally, we show item \eqref{item:C_0=C} of the definition (fullness of links), which we recall for the convenience of the reader:
 \begin{itemize}
  \item   If $v,w$ are distinct non-adjacent vertices of $\link(\Delta)$, for some simplex $\Delta$ of $\CX$, contained in
$\CW$--adjacent maximal simplices, then they are contained in $\CW$--adjacent simplices of the form $\Delta\star\Delta'$.
 \end{itemize}

It suffices to consider simplices $\Delta$ of $\bXi$--and $\mathcal K$--type.  Indeed, in all other cases (see Lemma~\ref{L:remaining simplices}), the vertices $v,w$ under consideration are contained in the link of a simplex $\Delta'$ containing $\Delta$ where $\Delta'$ is of $\bXi$--or $\mathcal K$--type (as can be seen by enlarging $\Delta$ until its link is no longer a join; $v$ and $w$ are not $\CX$--adjacent so they are contained in the same ``side'' of any join structure). Hence, once we deal with those cases, we know that there are suitable maximal simplices containing the larger simplex, whence $\Delta$.

  Consider first a simplex $\Delta$ of $\mathcal K$--type with vertices $s,v(s),w$. Consider distinct vertices $t_1,t_2$ (necessarily in $\kvtx$) of $\link(\Delta)$, and suppose that there are vertices $s_1,s_2 \in \kvtx$ so that the maximal simplices $\sigma(s_1,t_1)$ and $\sigma(s_2,t_2)$ are connected in $\CW$. There are two possibilities:

 \begin{itemize}
  \item $\bar d(M(s_1,t_1),M(s_2,t_2))\leq 10 R$. In this case, we have $\bar d(M(t_1),M(t_2))\leq 10R$.
    In particular, in view of the second bullet in the definition of the edges of $\CW$, we have that $t_1,t_2$ are contained, respectively, in the $\CW$--connected maximal simplices $\Delta*t_1=\sigma(s,t_1)$ and $\Delta* t_2=\sigma(s,t_2)$.
  \item $s_1=s_2$ and $\bar d(M(t_1),M(t_2))\leq 10R$ (notice that $t_1\neq t_2$ so that the ``symmetric'' case cannot occur). Again, we reach the same conclusion as above.
   \end{itemize}

 We can now consider a simplex $\Delta$ of $\bXi$--type with vertices $s,v(s)$. Consider vertices $x_1,x_2$ of $\link(\Delta)$ that are not $\CX$--adjacent but are contained in $\CW$--adjacent maximal simplices. Furthermore, we can assume that $x_1,x_2$ are not in the link of a simplex of $\mathcal K$--type (the case we just dealt with) which contains $\Delta$, since in that case we already know that there are suitable maximal simplices containing the larger simplex, whence $\Delta$. Then, using the structure of $\link(\Delta)$, we see that there are vertices $s_i,t_i \in \kvtx$ so that:
 \begin{itemize}
  \item  $x_i\in \{t_i,v(t_i)\}$,
  \item $t_i$ and $v(t_i)$ all belong to $\link(\Delta)$, and
    \item $\sigma(s_1,t_1)$ and $\sigma(s_2,t_2)$ are connected in $\CW$.
 \end{itemize}

 In turn, the last bullet splits into two cases:

 \begin{itemize}
  \item $\bar d(M(s_1,t_1),M(s_2,t_2))\leq 10 R$. In this we have $\bar d(M(t_1),M(t_2))\leq 10 R$, so that $x_1,x_2$ are contained, respectively, in the $\CW$--connected maximal simplices $\Delta*\{t_1, v(t_1)\}=\sigma(s,t_1)$ and $\Delta*\{t_2, v(t_2)\}=\sigma(s,t_2)$, so this case is fine.
  \item $s_1=s_2$ and $\bar d(M(t_1),M(t_2))\leq 10 R$. But again we reach the same conclusion as before.
 \end{itemize}
\end{proof}

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