text:
\begin{document}

\title[Ramanujan triangle complexes]{Spectrum and combinatorics of Ramanujan\\
triangle complexes}

\author{Konstantin Golubev and Ori Parzanchevski}
\begin{abstract}
Ramanujan graphs have extremal spectral properties, which imply a
remarkable combinatorial behavior. In this paper we compute the high
dimensional Hodge-Laplace spectrum of Ramanujan triangle complexes,
and show that it implies a combinatorial expansion property, and a
pseudo-randomness result. For this purpose we prove a Cheeger-type
inequality and a mixing lemma of independent interest.
\end{abstract}

\date{\today}

\maketitle
\tableofcontents{}

\section{\label{sec:Introduction}Introduction}

A $k$-regular graph is said to be \emph{Ramanujan} if its nontrivial
Laplace spectrum (see \prettyref{eq:graph-laplace}) is contained
in the interval $\mathcal{I}_{k}=\left[k-2\sqrt{k-1},k+2\sqrt{k-1}\right]$.
The reason for these particular values is the following theorem, which
says that asymptotically, they are the best that one can hope for:
\begin{thm*}[Alon-Boppana, \cite{LPS88,Nil91,grigorchuk1999asymptotic,li2001negative}]
Let $\left\{ G_{i}\right\} $ be a collection of $k$-regular graphs,
$\Spec\left(G_{i}\right)$ the nontrivial Laplace spectrum of $G_{i}$,
and $\lambda\left(G_{i}\right)$, $\Lambda\left(G_{i}\right)$ the
minimum and maximum of $\Spec\left(G_{i}\right)$, respectively.
\begin{enumerate}
\item If there are infinitely many different $G_{i}$ then $\liminf_{i}\lambda\left(G_{i}\right)\leq k-2\sqrt{k-1}$.
\item If $\left\{ \mathrm{girth}\left(G_{i}\right)\right\} $ is unbounded
then the closure of $\bigcup_{i}\Spec\left(G_{i}\right)$ contains
$\mathcal{I}_{k}$, and in particular $\limsup_{i}\Lambda\left(G_{i}\right)\geq k+2\sqrt{k-1}$.
\end{enumerate}
\end{thm*}
Bounds on the Laplace spectrum of a graph are extremely useful. The
so-called ``discrete Cheeger inequalities'' relate $\lambda\left(G\right)$
to combinatorial expansion, and the ``expander mixing lemma'' connects
spectral concentration (namely $\Lambda\left(G\right)-\lambda\left(G\right)$)
to pseudo-randomness. Thus, Ramanujan graphs are excellent \emph{expanders}:
we refer to \cite{HLW06,Tao11,Lub12} for detailed surveys on expander
graphs, their properties and applications.

Ramanujan graphs were first constructed in the works \cite{LPS88,margulis1988explicit},
using the Ramanujan\textendash{}Petersson conjecture for $\mathrm{GL}_{2}$,
which was proved by Eichler and Deligne in characteristic zero, and
by Drinfeld in positive characteristic (the latter being used to construct
Ramanujan graphs in \cite{morgenstern1994existence}). The constructed
graphs are certain quotients of the Bruhat-Tits building associated
with $\mathrm{PGL}_{2}$ over a nonarchimedean local field. This building
is a $k$-regular tree, and this gives the real reason for the values
which appear in the Alon-Boppana theorem: the interval $\mathcal{I}_{k}$
is precisely the $L^{2}$-spectrum of the Laplacian on a $k$-regular
tree, by a classic result of Kesten \cite{Kesten1959}. Thus, Ramanujan
graphs are those whose nontrivial spectrum is contained in that of
their universal cover: this relation is a discrete analogue of the
original Ramanujan conjecture (which regards quotients of the hyperbolic
plane), and this is what led Lubotzky, Phillips and Sarnak to the
name \emph{Ramanujan graphs}. For the complete story of Ramanujan
graphs we refer the reader to the monographs \cite{Sarnak1990,Lub10}.

In recent years, several authors turned to study higher dimensional
Ramanujan complexes, which are quotients of the Bruhat-Tits buildings
associated with $\mathrm{PGL}_{d}$. This includes the works \cite{ballantine2000ramanujan,cartwright2003ramanujan,li2004ramanujan,Lubotzky2005a,Lubotzky2005b,sarveniazi2007explicit,kang2010zeta,kang2014zeta},
and a recent survey of the field appears in \cite{Lubotzky2013}.
This surge of interest was prompted by advances in the theory of automorphic
representations, especially the resolution of the generalized Ramanujan
conjecture for $\mathrm{GL}_{d}$ in positive characteristic by Lafforgue
\cite{lafforgue2002chtoucas}. While Ramanujan complexes form natural
generalizations of Ramanujan graphs, it is not clear what are the
best generalizations of graph theoretic notions such as Laplace spectrum,
Cheeger constant, pseudo-randomness, girth and so on. Until now, most
of the research has focused on the spectral theory of the so-called
\emph{Hecke operators}, which act again on the vertices of the complex,
and sum up to the graph adjacency operator. Our interest lies in the
the spectrum of the high-dimensional Hodge-Laplace operators, also
known as combinatorial Laplacians. These originate in \cite{Eck44},
and form natural analogues of the Hodge-Laplace operators acting on
the spaces of differential forms on a Riemannian manifold. The $i$-th
simplicial Laplacian, denoted $\Delta_{i}^{+}$, acts on the simplicial
$i$-forms of the complex, which are skew-symmetric functions on oriented
cells of dimension $i$. The $0$-forms are just the functions on
vertices, and $\Delta_{0}^{+}$ is the standard graph Laplacian.

An Alon-Boppana theorem holds here as well:
\begin{thm}[\cite{li2004ramanujan,Parzanchevski2012walks}]
\label{thm:Alon-Boppana-High}If $\left\{ X_{i}\right\} $ is a family
of quotients of the Bruhat-Tits building $\mathcal{B}_{d}$ with unbounded
injectivity radius
\footnote{Note that the injectivity radius of a quotient of a tree is half its
girth.
}, then the closure of $\bigcup_{i}\Spec\Delta_{j}^{+}\left(X_{i}\right)$
contains the $L^{2}$-spectrum of $\Delta_{j}^{+}\left(\mathcal{B}_{d}\right)$.
\end{thm}
(The statement in \cite{li2004ramanujan} addresses the simultaneous
spectrum of the aforementioned Hecke operators, and this implies the
theorem for $\Delta_{0}^{+}$; \cite{Parzanchevski2012walks} handles
the Laplacians in higher dimensions. For a different generalization
of the Alon-Boppana theorem see also \cite{First2014}.)

As in graphs, the Ramanujan quotients of $\mathcal{B}_{d}$ should
be those whose Laplace spectrum, in every dimension, is contained
within the corresponding $L^{2}$-spectrum of $\mathcal{B}_{d}$.
But what is the spectrum of $\Delta_{i}^{+}$ on $\mathcal{B}_{d}$
and its Ramanujan quotients? It is well known that the spectrum of
$\Delta_{0}^{+}$ (i.e., the underlying graph spectrum) is concentrated,
though not as good as in Ramanujan graphs \cite{macdonald1979symmetric,li2004ramanujan,Lubotzky2005a}.
For higher dimensions, a celebrated local-to-global argument of Garland
\cite{Gar73} establishes lower bounds on the $\Delta_{i}^{+}$-spectra
(for any complex, not necessarily $\mathcal{B}_{d}$ and its quotients).
While Garland's argument can be adjusted to give an upper bound as
well (see \cite{papikian2008eigenvalues,Gundert2012}), the local
picture of $\mathcal{B}_{d}$ gives trivial bounds for the higher
end of the spectra of its quotients.

The purpose of this paper is twofold: to compute the higher Laplace
spectrum of Ramanujan triangle complexes, and to relate the simplicial
Laplace spectra to combinatorial properties of the complex. The first
half of the paper (\prettyref{sec:Iwahori-Hecke-boundary-maps}-\prettyref{sec:Unitary-Iwahori-spherical-repres})
is dedicated to \prettyref{thm:ram-triangle-spec}, which determines
explicitly the spectrum of $\Delta_{1}^{+}$ for Ramanujan triangle
complexes. We find that the picture for $\Delta_{1}^{+}$ is more
intricate than that in dimension zero: most of the spectrum is concentrated
in the strip $\mathcal{I}_{k_{1}}=\left[k_{1}-2\sqrt{k_{1}-1},k_{1}+2\sqrt{k_{1}-1}\right]$,
where $k_{1}$ is the degree of edges in the complex. However, a small
fraction of the spectrum is concentrated within the narrow strip $\left[2k_{1}-1,2k_{1}+8\right]$
(see \prettyref{thm:ram-triangle-spec}). This gives, in particular,
an explanation for the failure of Garland's method to achieve spectral
concentration.

Our computational approach is inspired by \cite{Lub10,Lubotzky2005a},
and especially \cite{kang2010zeta}, but it is different: rather then
computing the Laplacians directly, we show (\prettyref{prop:Iwahori-boundaries})
that certain elements of the Iwahori-Hecke algebra of $\mathrm{PGL}_{3}$
act on appropriate representations as simplicial boundary and coboundary
maps (see \prettyref{sec:Complexes-and-buildings}). We then invoke
the classification of unitary, Iwahori-spherical representations of
$\mathrm{PGL}_{3}$ which is computed explicitly in \cite{kang2010zeta}
(using results from \cite{borel1976admissible,zelevinsky1980induced,casselman1980unramified,tadic1986classification}),
to obtain the complete spectrum. While much of the work should be
useful for higher dimensions as well, the relevant classification
problem becomes harder and we do not pursue it here.

\bigskip{}

As in the case of Ramanujan graphs, we would like to infer combinatorial
properties from the spectral theory of the complex, and the second
half of the paper (\prettyref{sec:comb-expansion} and \prettyref{sec:pseudo-randomness})
is devoted to this purpose. It can be read independently from \prettyref{sec:Iwahori-Hecke-boundary-maps}-\prettyref{sec:Unitary-Iwahori-spherical-repres},
as the theorems therein apply to arbitrary complexes and not only
to quotients of Bruhat-Tits buildings.

In \prettyref{sec:comb-expansion} we prove a Cheeger-type inequality,
which uses the concentration of the vertex spectrum, and the fact
that the edge spectrum is bounded away from zero:
\begin{thm}
\label{thm:Cheeger}Let $X$ be a triangle complex on $n$ vertices,
with nontrivial $\Delta_{0}^{+}$-spectrum contained in $\left[k_{0}-\mu_{0},k_{0}+\mu_{0}\right]$,
and nontrivial $\Delta_{1}^{+}$-spectrum bounded below by $\lambda_{1}$.
Then for any partition of the vertices of $X$ into nonempty sets
\textup{$A,B,C$, one has
\[
\frac{\left|T\left(A,B,C\right)\right|n^{2}}{\left|A\right|\left|B\right|\left|C\right|}\geq\lambda_{1}\left(k_{0}-\mu_{0}\left(1+\frac{10\, n^{3}}{9\left|A\right|\left|B\right|\left|C\right|}\right)\right),
\]
}where $T\left(A,B,C\right)$ is the set of triangles with one vertex
in each of $A,B$ and $C$.
\end{thm}
For complexes with a complete skeleton (underlying graph) this inequality
reduces to the one which appears in \cite{parzanchevski2012isoperimetric}.
It can also be generalized to higher dimensions - this is done in
\prettyref{thm:cheeger-gen-dim}. For an interpretation in terms of
a Cheeger-type constant, see \prettyref{eq:cheeger-triangle-def}
and \prettyref{eq:cheeger-ineq-triangle}.

In \prettyref{sec:pseudo-randomness} we turn to pseudo-randomness,
and use the full strength of \prettyref{thm:ram-triangle-spec}. In
a previous paper \cite{Parzanchevski2013} we have shown that simultaneous
concentration of both vertex and edge spectra implies pseudo-randomness,
in the sense that $\left|T\left(A,B,C\right)\right|$ is close to
its expected value for any large disjoint sets $A,B,C$ (not necessarily
partitions). Since in Ramanujan triangle complexes the edge spectrum
is not concentrated, having eigenvalues near both $k_{1}$ and $2k_{1}$,
this result cannot be applied directly. Instead, we use the concentration
around these two values to deduce a pseudo-random behavior of \emph{$2$-galleries},
namely, pairs of triangles which intersect in an edge. We denote by
$F^{2}\left(A,B,C,D\right)$ the galleries $\left(t,t'\right)$ such
that $t\in T\left(A,B,C\right)$ and $t'\in T\left(B,C,D\right)$
(see Figure \ref{fig:2-gallery}).
\begin{thm}
\label{thm:Pseudo-random}Let $X$ be a tripartite Ramanujan triangle
complex on $n$ vertices with edge degrees $k_{1}=q+1$. Let $A,B,C,D$
be disjoint sets of vertices such that each of $A\cup D$, $B$ and
$C$ is contained in a different block of the three-partition of $X$.
If $A,B,C$ and $D$ are of sizes at most $\vartheta n$, then
\[
\left|\left|F^{2}\left(A,B,C,D\right)\right|-\frac{27q^{4}\left|A\right|\left|B\right|\left|C\right|\left|D\right|}{n^{3}}\right|\leq\left(65q^{3.5}\vartheta+244q^{2.5}\right)\vartheta n.
\]

\end{thm}
A version which applies to general triangle complexes appears in \prettyref{thm:mixing-general}.
As an application of the theorem we show that the weak chromatic number
of non-tripartite Ramanujan triangle complexes is $\Omega\left(\sqrt[3]{q}\right)$,
improving on the bound $\Omega\left(\sqrt[6]{q}\right)$ which is
obtained in \cite{evra2014mixing}.

\begin{figure}[h]
\noindent \noindent \captionsetup{format=plain, margin=1.5em, justification=raggedright}
\floatbox[{\capbeside}]{figure} {\caption{A $2$-gallery through $A$, $B$, $C$, $D$ in a tripartite triangle complex.}\label{fig:2-gallery}} {\includegraphics[scale=0.65]{gallery.pdf}}
\end{figure}

\begin{acknowledgement*}
We would like to thank Alex Lubotzky for providing us with guidance
and inspiration, and to Uriya First and Dima Trushin for patiently
introducing us to the representation theory of $p$-adic groups. We
are grateful to Shai Evra, Alex Kontorovich, Winnie Li, Doron Puder
and Peter Sarnak for helpful discussions and suggestions. The first
author was supported by the ERC and by the Israel Science Foundation.
The second author was supported by The Fund for Math and by NSF grant
DMS-1128155, and is grateful for the hospitality of the Institute
for Advanced Study.
\end{acknowledgement*}

\section{\label{sec:Complexes-and-buildings}Complexes and buildings}

In this section we recall the basic elements of simplicial Hodge theory,
and the definition of affine Bruhat-Tits buildings. For a more relaxed
exposition of the former we refer to \cite[§2]{parzanchevski2012isoperimetric},
and for the latter to \cite{li2004ramanujan,Lubotzky2005a,Lubotzky2013}.

Let $X$ be a finite simplicial complex of dimension $d$, and denote
by $X^{i}$ the set of (unoriented) cells of dimension $i$ in $X$
($i$\emph{-cells}). The \emph{degree} of an $i$-cell is the number
of $\left(i+1\right)$-cells which contain it. We denote by $\Omega^{i}=\Omega^{i}\left(X\right)$
the space of $i$-forms, namely skew-symmetric complex functions on
the oriented $i$-cells, equipped with the inner product $\left\langle f,g\right\rangle =\sum_{\sigma\in X^{i}}f\left(\sigma\right)\overline{g\left(\sigma\right)}$.
The $i$-th boundary map $\partial_{i}:\Omega^{i}\rightarrow\Omega^{i-1}$
is defined by $\left(\partial_{i}f\right)\left(\sigma\right)=\sum_{v:v\sigma\in X^{i}}f\left(v\sigma\right)$,
and its dual is the $i$-th coboundary map $\delta_{i}=\partial_{i}^{*}:\Omega^{i-1}\rightarrow\Omega^{i}$,
given by $\left(\delta_{i}f\right)\left(\sigma\right)=\sum_{j=0}^{i}\left(-1\right)^{j}f\left(\sigma\backslash\sigma_{j}\right)$.
We use the standard notation $Z_{i}=\ker\partial_{i}$, $Z^{i}=\ker\delta_{i+1}$,
$B_{i}=\im\partial_{i+1}$ and $B^{i}=\im\delta_{i}$ for cycles,
cocycles, boundaries and coboundaries.

The \emph{upper, lower} and \emph{full $i$-Laplacians} are $\Delta_{i}^{+}=\partial_{i+1}\delta_{i+1}$,
$\Delta_{i}^{-}=\delta_{i}\partial_{i}$ and $\Delta_{i}=\Delta_{i}^{+}+\Delta_{i}^{-}$,
respectively. For our purposes the upper Laplacian is the most important
one, and as we have remarked, $\Delta_{0}^{+}$ is the classical Laplacian
of graph theory:
\begin{equation}
\left(\Delta_{0}^{+}f\right)\left(v\right)=\deg\left(v\right)f\left(v\right)-\sum_{w\sim v}f\left(w\right).\label{eq:graph-laplace}
\end{equation}
Since $\left(\Omega^{*},\delta_{*}\right)$ is a cochain complex,
$\im\delta_{i}$ is always in $\ker\Delta_{i}^{+}$, and the corresponding
zeros in $\Spec\Delta_{i}^{+}$ are considered the \emph{trivial spectrum}
(of $\Delta_{i}^{+}$). The nontrivial $i$-spectrum is thus
\begin{equation}
\Spec\Delta_{i}^{+}\big|_{\left(\im\delta_{i}\right)^{\bot}}=\Spec\Delta_{i}^{+}\big|_{\ker\partial_{i}}=\Spec\Delta_{i}^{+}\big|_{Z_{i}},\label{eq:nontrivial-spec}
\end{equation}
and zero is in the nontrivial spectrum if and only if the $i$-th
cohomology of $X$ (over $\mathbb{C}$) is nontrivial. We should remark
that in the theory of Ramanujan graphs, the Laplacian eigenvalue $2k$
(for a $k$-regular graph) is also considered trivial. We will return
to this point later.

Next, let $F$ be a nonarchimedean local field with ring of integers
$\mathcal{O}$, uniformizer $\pi$, and residue field $\nicefrac{\mathcal{O}}{\pi\mathcal{O}}$
of size $q$, which we shall identify with $\mathbb{F}_{q}$. The
canonical examples are $F=\mathbb{Q}_{p}$ with $\mathcal{O}=\mathbb{Z}_{p}$
and $\pi=q=p$, and $F=\mathbb{F}_{p^{e}}\left(\left(t\right)\right)$
with $\mathcal{O}=\mathbb{F}_{p^{e}}\left[\left[t\right]\right]$,
$\pi=t$ and $q=p^{e}$. The affine Bruhat-Tits building $\mathcal{B}=\mathcal{B}_{d}\left(F\right)$
associated with $\mathrm{PGL}_{d}\left(F\right)$ is an infinite contractible
complex of dimension $d-1$. Denoting $G=\mathrm{PGL}_{d}\left(F\right)$
and $K=\mathrm{PGL}_{d}\left(\mathcal{O}\right)$, the vertices of
$\mathcal{B}$ are in correspondence with the left $K$-cosets in
$G$. To every vertex $gK$ we associate the $\mathcal{O}$-lattice
$g\mathcal{O}^{d}$, which is only defined up to scaling by $F^{\times}$,
since $g\in\mathrm{PGL}_{d}\left(F\right)=\nicefrac{\mathrm{GL}_{d}\left(F\right)}{F^{\times}}$.
A collection of vertices $\left\{ g_{i}K\right\} _{i=0..r}$ forms
an $r$-cell if, possibly after reordering, there exist representatives
$g'_{i}\in\mathrm{GL}_{d}\left(F\right)$ for $g_{i}$, such that
the corresponding lattices satisfy
\begin{equation}
\pi g'_{0}\mathcal{O}^{d}<g'_{r}\mathcal{O}^{d}<g'_{r-1}\mathcal{O}^{d}<\ldots<g'_{1}\mathcal{O}^{d}<g'_{0}\mathcal{O}^{d}.\label{eq:cell-relation}
\end{equation}
This relation turns out to be equivalent to having an edge between
every $g_{i}K$ and $g_{j}K$ (i.e., representatives $g_{i}',g_{j}'\in\mathrm{GL}_{d}\left(F\right)$
with $\pi g_{i}^{'}\mathcal{O}^{d}<g_{j}'\mathcal{O}^{d}<g_{i}'\mathcal{O}^{d}$).
In other words, $\mathcal{B}$ is a clique (or ``flag'') complex.

The action of $G$ on $\nicefrac{G}{K}$ defines an action on the
complex $\mathcal{B}$, as it clearly preserves \prettyref{eq:cell-relation}.
Given a cocompact, torsion-free lattice $\Gamma\leq G$, the quotient
$\Gamma\backslash\mathcal{B}$ is a finite complex. The building $\mathcal{B}_{2}$
is a $\left(q+1\right)$-regular tree, and its quotients by lattices
in $G$ are $\left(q+1\right)$-regular graphs. Certain lattices give
rise to Ramanujan quotients:
\begin{thm*}[\cite{LPS88,margulis1988explicit}, cf.\ \cite{Sarnak1990,Lub10}]
If $\Gamma$ is a congruence subgroup of a torsion-free arithmetic
lattice in $G$, then $\Gamma\backslash\mathcal{B}_{2}$ is a Ramanujan
graph.
\end{thm*}
As we have remarked, some of these graphs have the eigenvalue $2k=2\left(q+1\right)$
in their spectrum (with the rest of the nontrivial spectrum in $\mathcal{I}_{k}$).
This is the origin of this eigenvalue: the vertices of $\mathcal{B}_{d}$
admit a coloring in $\nicefrac{\mathbb{Z}}{d\mathbb{Z}}$, defined
by $\col\left(gK\right)=\ord_{\pi}\left(\det g\right)+d\mathbb{Z}$,
but the action of $G$ does not respect the coloring, since $\col\left(gv\right)=\col\left(v\right)+\ord_{\pi}\left(\det g\right)$.
The following are equivalent:
\begin{itemize}
\item $2k=2\left(q+1\right)$ appears in $\Spec\Delta_{0}^{+}\left(\Gamma\backslash\mathcal{B}_{2}\right)$.
\item The action of $\Gamma$ preserves the color of vertices.
\item $\col\Gamma\equiv0$ (i.e.\ $\ord_{\pi}\left(\det\gamma\right)\in2\mathbb{Z}$
for all $\gamma\in\Gamma$).
\item $\Gamma\backslash\mathcal{B}_{2}$ is bipartite.
\end{itemize}
Together with \cite{morgenstern1994existence}, the papers \cite{LPS88,margulis1988explicit}
establish that if $k-1$ is a prime power, then there exist infinitely
many non-bipartite $k$-regular Ramanujan graphs, and infinitely many
bipartite ones. For general $k$, the existence of infinitely many
bipartite Ramanujan graphs was only settled in \cite{marcus2013interlacing}
(by a non-constructive argument), and it is still open for the non-bipartite
case.

\medskip{}

We now continue to dimension two. The Bruhat-Tits building $\mathcal{B}_{3}$
is a regular triangle complex, with vertex and edge degrees
\[
k_{0}=2\left(q^{2}+q+1\right)\qquad\mbox{and}\qquad k_{1}=q+1
\]
respectively. There are several plausible ways to define what are
the Ramanujan quotients of $\mathcal{B}_{d}$, and these are discussed
in \cite{cartwright2003ramanujan,li2004ramanujan,Lubotzky2005a,kang2010zeta,First2014}.
Luckily, they all coincide for $\mathrm{PGL}_{3}$ (see \cite{kang2010zeta}),
though not in general (see \cite{fang2013zeta}). The definition we
use is the following (the notions in the definition are explained
throughout \prettyref{sec:Iwahori-Hecke-boundary-maps}):
\begin{defn}
A quotient of $\mathcal{B}_{3}$ by a torsion-free cocompact lattice
$\Gamma$ in $G=\mathrm{PGL}_{3}\left(F\right)$ is \emph{Ramanujan}
if every Iwahori-spherical representation of $G$ which appears in
$L^{2}\left(\Gamma\backslash G\right)$ is either finite-dimensional
or tempered.
\end{defn}
The papers \cite{li2004ramanujan,Lubotzky2005b,sarveniazi2007explicit}
give several constructions of Ramanujan complexes, some of which are
the clique complexes of Cayley graphs. \medskip{}

We begin the study of these complexes at the high end of the spectrum.
It is not hard to see that a $d$-complex whose $j$-cells have degrees
bounded by $k_{j}$ satisfies $\Spec\Delta_{j}^{+}\subseteq\left[0,\left(j+2\right)k_{j}\right]$
(see e.g.\ \cite{parzanchevski2012isoperimetric}). We say that the
complex is $\left(d+1\right)$\emph{-partite} if there exists a $\left(d+1\right)$-coloring
of its vertices such that no two vertices in a $d$-cell are of the
same color. If $X$ is a $\left(d+1\right)$-partite $d$-complex
whose $\left(d-1\right)$-cells have degree $k_{d-1}$, then $\left(d+1\right)k_{d-1}\in\Spec\Delta_{d-1}^{+}$.
Unlike the graph case, the converse may fail: rather than $\left(d+1\right)$-partiteness,
the $\left(d+1\right)k_{d-1}$ eigenvalue indicates \emph{disorientability}
(see \cite{Parzanchevski2012walks}), the existence of an orientation
of the $d$-cells so that they \emph{agree} on the orientation of
their codimension one intersections.

In our case, all quotients of $\mathcal{B}_{3}$ by a lattice in $G$
are disorientable, i.e.\ have $3k_{1}\in\Spec\Delta_{1}^{+}$, and
a similar situation holds in higher dimensions (see \cite{papikian2008eigenvalues}).
The reason is the following: if we define the color of a directed
edge to be $\col\left(\left[v,w\right]\right)=\col\left(w\right)-\col\left(v\right)$
(in $\nicefrac{\mathbb{Z}}{d\mathbb{Z}}$), then the action of $G$
does preserve edge colors, since $\col\left(gv\right)=\col\left(v\right)+\ord_{\pi}\left(\det g\right)$.
Thus, the disorientation eigenform $f\left(e\right)=\left(-1\right)^{\col e}$
always factors modulo $\Gamma$, giving $3k_{1}\in\Spec\Delta_{1}^{+}$.

If $\Gamma$ also preserves vertex coloring, i.e.\ $\col\Gamma\equiv0$,
then $X=\Gamma\backslash\mathcal{B}_{3}$ is naturally tripartite.
The converse also holds: if $\col\Gamma\not\equiv0$ then
\begin{equation}
\widehat{\Gamma}=\left\{ \gamma\in\Gamma\,\middle|\,\col\gamma K=0\right\} =\Gamma\cap\ker\left(\ord_{\pi}\det:G\rightarrow\nicefrac{\mathbb{Z}}{3\mathbb{Z}}\right)\label{eq:Gamma-hat}
\end{equation}
is a normal subgroup of $\Gamma$ of index three, and $\widehat{X}=\widehat{\Gamma}\backslash\mathcal{B}_{3}$
is a tripartite three-cover of $X$, with the coloring $\col=\ord_{\pi}\det$.
But any two vertices in $\mathcal{B}_{3}$, and thus in $\widehat{X}$,
can be connected by a $2$-gallery, that is, a sequence of triangles
in which every consecutive pair intersect in an edge. This shows that
$\widehat{X}$ admits a unique $3$-coloring (up to renaming of colors).
Thus, any three-coloring of $X$ lifts to $\col$, which must therefore
be $\Gamma$-invariant, contradicting $\col\Gamma\not\equiv0$.

Having understood the ``colored'' part of the spectrum
\footnote{We reserve the term ``trivial'' for the homological meaning - zeros
obtained on coboundaries.
}, we can move on to its interesting parts:
\begin{thm}
\label{thm:ram-triangle-spec}Let $X$ be a non-tripartite Ramanujan
triangle complex on $n$ vertices.
\begin{enumerate}
\item The nontrivial spectrum of $\Delta_{0}^{+}$ is contained within $\left[k_{0}-6q,k_{0}+3q\right]$.
\item The nontrivial spectrum of $\Delta_{1}^{+}$ consists of:

\begin{enumerate}
\item $n\left(q^{2}+q-2\right)+2$ eigenvalues in the strip
\begin{equation}
\mathcal{I}=\left[k_{1}-2\sqrt{q},k_{1}+2\sqrt{q}\right].\label{eq:excellent-interval}
\end{equation}

\item For every nontrivial $\lambda\in\Spec\Delta_{0}^{+}$, the eigenvalues
$\frac{3k_{1}}{2}\pm\sqrt{\left(\frac{3k_{1}}{2}\right)^{2}-\lambda}$.
\\
This amounts to $n-1$ eigenvalues in each of the strips
\begin{equation}
\begin{aligned}\mathcal{I}_{-} & =\left[\tfrac{3k_{1}}{2}-\sqrt{\left(\tfrac{k_{1}}{2}\right)^{2}+8q},k_{1}+1\right]\\
\mathcal{I}_{+} & =\left[2k_{1}-1,\tfrac{3k_{1}}{2}+\sqrt{\left(\tfrac{k_{1}}{2}\right)^{2}+8q}\right].
\end{aligned}
\label{eq:good-bad-interval}
\end{equation}

\item The disorientation eigenvalue $3k_{1}$, corresponding to $f\left(e\right)=\left(-1\right)^{\col e}$.
\end{enumerate}
\end{enumerate}
If $X$ is tripartite (and Ramanujan):
\begin{enumerate}
\item The nontrivial spectrum of $\Delta_{0}^{+}$ consists of:

\begin{enumerate}
\item $n-3$ eigenvalues of $\Delta_{0}^{+}$ in $\left[k_{0}-6q,k_{0}+3q\right]$.
\item The eigenvalue $\frac{3k_{0}}{2}$, twice, corresponding to $f\left(v\right)=\exp\left(\pm\frac{2\pi i}{3}\col\left(v\right)\right)$.
\end{enumerate}
\item The nontrivial spectrum of $\Delta_{1}^{+}$ consists of:

\begin{enumerate}
\item $n\left(q^{2}+q-2\right)+6$ eigenvalues in $\mathcal{I}$.
\item $n-3$ eigenvalues in each of $\mathcal{I}_{\pm}$, corresponding
to $\frac{3k_{1}}{2}\pm\sqrt{\left(\frac{3k_{1}}{2}\right)^{2}-\lambda}$
for the eigenvalues of $\Delta_{0}^{+}$ in $\left(a\right)$ above.
\item As before, $3k_{1}$.
\end{enumerate}
\end{enumerate}
\end{thm}
As we have mentioned, the spectrum of $\Delta_{0}^{+}$ is well-known
\cite{macdonald1979symmetric,li2004ramanujan,Lubotzky2005a}, but
our computation methods are different, and give the spectrum in both
dimensions in a unified manner. Let us make a few remarks:
\begin{enumerate}
\item These bounds cannot be improved: by \prettyref{thm:Alon-Boppana-High},
a sequence of quotients with unbounded injectivity radius (as constructed
in \cite{lubotzky2007moore}) has Laplace spectra which accumulate
to any point in these intervals.
\item The strip $\mathcal{I}_{-}$ is contained in $\mathcal{I}$, and both
of the strips $\mathcal{I}_{\pm}$ are highly concentrated:
\[
\mathcal{I}_{-}\subseteq\left[k_{1}-8,k_{1}+1\right],\qquad\mathcal{I}_{+}\subseteq\left[2k_{1}-1,2k_{1}+8\right],
\]
so that in particular
\[
\Spec\Delta_{1}^{+}\big|_{Z_{1}}\subseteq\left[k_{1}-2\sqrt{k_{1}-1},k_{1}+2\sqrt{k_{1}-1}\right]\cup\left[2k_{1}-1,2k_{1}+8\right]\cup\left\{ 3k_{1}\right\} .
\]

\item There are no nontrivial zeros in the spectra of $\Delta_{0}^{+}$
and $\Delta_{1}^{+}$, so that the zeroth and first (torsion-free)
homologies of $X$ vanish, in accordance with \cite{Gar73,Casselman1974}.
\end{enumerate}
It is interesting to compare the theorem with Garland's (extended)
method, which relates the high-dimensional Laplace spectrum to the
graph spectrum of links of cells in codimension two. We denote by
$\lk\sigma$ the link of a cell $\sigma$.
\begin{thm*}[{\cite{Gar73,papikian2008eigenvalues,Gundert2012}
\footnote{The cited papers incorporate normalized versions of the Laplacian.
The version we use follows from \cite[Lem.\ 4.2]{parzanchevski2012isoperimetric}.
}}]
If $X$ is a finite complex such that $\Spec\Delta_{0}^{+}\left(\lk\sigma\right)\big|_{Z_{0}\left(\lk\sigma\right)}\subseteq\left[\lambda,\Lambda\right]$
for every $\sigma\in X^{j-2}$ and $k\leq\deg\sigma\leq K$ for every
$\sigma\in X^{j-1}$, then
\[
\Spec\Delta_{j}^{+}\left(X\right)\big|_{Z_{j}\left(X\right)}\subseteq\left[\left(j+1\right)\lambda-jK,\left(j+1\right)\Lambda-jk\right].
\]

\end{thm*}
The link of every vertex in $\mathcal{B}_{3}$ is the incidence graph
of the projective plane over $\mathbb{F}_{q}$. This is a bipartite
$k_{1}$-regular graph with
\[
\Spec\Delta_{0}^{+}\big|_{Z_{0}}=\left\{ k_{1}-\sqrt{q},k_{1}+\sqrt{q},2k_{1}\right\} ,
\]
and therefore by Garland any quotient of $\mathcal{B}_{3}$ satisfies
\[
\Spec\Delta_{1}^{+}\big|_{Z_{1}}\subseteq\left[k_{1}-2\sqrt{q},3k_{1}\right].
\]
\prettyref{thm:ram-triangle-spec} shows that both ends are tight!
(for the high end this follows from disorientability, and was already
observed in \cite{papikian2008eigenvalues}.) The problem with Garland's
method is that it completely misses the sparse picture within this
interval, which is crucial for our combinatorial purposes. We return
to the concentration of the spectrum in \prettyref{sec:comb-expansion}
and \prettyref{sec:pseudo-randomness}. The proof of \prettyref{thm:ram-triangle-spec}
occupies the next three sections.

\section{\label{sec:Iwahori-Hecke-boundary-maps}Iwahori-Hecke boundary maps}

Let us fix the fundamental vertex $v_{0}=K$ in $\mathcal{B}^{0}=\nicefrac{G}{K}$.
A torsion-free lattice $\Gamma\leq G$ acts freely on $\mathcal{B}^{0}$,
and for a quotient $X=\Gamma\backslash\mathcal{B}$ we denote by $v$
the image of $v\in\mathcal{B}^{0}$ in $X^{0}$ as well. The vertices
of $X$ are in correspondence with the double cosets in $\Gamma\backslash G/K$.
If $g_{1},\ldots,g_{n}$ are representatives for $\Gamma\backslash G/K$,
then $D=\coprod_{i=1}^{n}g_{i}K$ is a fundamental domain for $\Gamma$
in $G$; indeed, $\gamma D\cap D\neq\varnothing$ implies $\gamma g_{i}K\cap g_{j}K\neq\varnothing$
for some $i,j$, forcing $i=j$, and thus $\gamma\in g_{i}Kg_{i}^{-1}$;
but $\Gamma$ acts freely on $\mathcal{B}^{0}$ and $g_{i}Kg_{i}^{-1}$
fixes $g_{i}v_{0}$, so that $\gamma=1$. Thus, normalizing the Haar
measure on $G$ so that $\mu\left(K\right)=1$, we have $\mu\left(\Gamma\backslash G\right)=\mu\left(D\right)=n$.
Since $\Gamma$ does not intersect any conjugate of $K$, the measure
induced by $\mu$ on $\Gamma\backslash G/K$ is the counting measure,
so that there is a linear isometry $\Omega^{0}\left(X\right)\cong L^{2}\left(\Gamma\backslash G/K\right)$,
given explicitly by $f\left(gv_{0}\right)=f\left(\Gamma gK\right)$.
It will also be useful for us to identify $L^{2}\left(\Gamma\backslash G/K\right)$
with $L^{2}\left(\Gamma\backslash G\right)^{K}$, the space of $K$-fixed
vectors in the $G$-representation $L^{2}\left(\Gamma\backslash G\right)$,
in the natural manner.

The element $\sigma=\left(\begin{smallmatrix}0 & 1 & 0\\
0 & 0 & 1\\
\pi & 0 & 0
\end{smallmatrix}\right)\in G$ acts on $\mathcal{B}$ by rotation on the triangle consisting of
the vertices $v_{0}$, $\sigma v_{0}$, and $\sigma^{2}v_{0}$. We
choose as a fundamental oriented edge $e_{0}=\left[v_{0},\sigma v_{0}\right]$,
and denote
\begin{equation}
E=\stab_{G}e_{0}=K\cap\sigma K\sigma^{-1}=\left\{ \left(\begin{smallmatrix}* & * & *\\
* & * & *\\
x & y & *
\end{smallmatrix}\right)\in K\,\middle|\, x,y\in\pi\mathcal{O}\right\} \label{eq:E-def}
\end{equation}
($E$ is sometimes called a ``parahoric subgroup''.) Recall the
coloring $\col:\mathcal{B}^{1}\rightarrow\nicefrac{\mathbb{Z}}{3\mathbb{Z}}$
given by $\col\left(\left[gK,g'K\right]\right)=\ord_{\pi}\det\left(g'g^{-1}\right)$.
It is invariant under $G$, and thus well defined on $X^{1}=\Gamma\backslash\mathcal{B}^{1}$.
Every edge in $X$ (and $\mathcal{B}$) has one orientation with color
$1$, and one with color $2$ (there are no edges of color $0$).
Furthermore, $G$ acts transitively on the non-oriented edges of $\mathcal{B}$,
so that the oriented edges of color $1$ in $\mathcal{B}$ can be
identified with $ $$G/E$ (as $\col e_{0}=1$), and those of $X$
with $\Gamma\backslash G/E$, giving
\[
\mu\left(E\right)=\frac{\mu\left(\Gamma\backslash G\right)}{\left|\Gamma\backslash G/E\right|}=\frac{n}{\left|X^{1}\right|}=\frac{1}{q^{2}+q+1},
\]
and we can identify $\Omega^{1}\left(X\right)$ with $L^{2}\left(\Gamma\backslash G\right)^{E}$,
by
\begin{equation}
f\left(ge_{0}\right)=f\left(\left[gv_{0},g\sigma v_{0}\right]\right)=\sqrt{\mu\left(E\right)}f\left(\Gamma g\right),\quad f\left(\left[g\sigma v_{0},gv_{0}\right]\right)=-\sqrt{\mu\left(E\right)}f\left(\Gamma g\right).\label{eq:1-forms-as-E-stable-vecs}
\end{equation}

The scaling by $\sqrt{\mu\left(E\right)}$ is needed to make the isomorphism
$\Omega^{1}\left(X\right)\cong L^{2}\left(\Gamma\backslash G\right)^{E}$
an isometry: if $g_{1},\ldots,g_{nk_{0}/2}$ is a set of representatives
for $\Gamma\backslash G/E$, then for $f\in L^{2}\left(\Gamma\backslash G\right)^{E}$
we have
\begin{align*}
\left\Vert f\right\Vert _{\Omega^{1}\left(X\right)}^{2} & =\sum_{i=1}^{nk_{0}/2}\left|f\left(g_{i}e_{0}\right)\right|^{2}=\sum_{i=1}^{nk_{0}/2}\mu\left(E\right)\left|f\left(\Gamma g_{i}\right)\right|^{2}=\sum_{i=1}^{nk_{0}/2}\int_{E}\left|f\left(\Gamma g_{i}\right)\right|^{2}de\\
 & =\sum_{i=1}^{nk_{0}/2}\int_{E}\left|f\left(\Gamma g_{i}e\right)\right|^{2}de=\int_{\Gamma\backslash G}\left|f\left(\Gamma g\right)\right|^{2}dg=\left\Vert f\right\Vert _{L^{2}\left(\Gamma\backslash G\right)}^{2}.
\end{align*}
(Let us remark that this is simpler than the picture in higher dimensions:
for example, $G=\mathrm{PGL}_{4}\left(F\right)$ acts transitively
on the edges of color $1$, and on the edges of color $3$ in $\mathcal{B}_{4}$.
It also preserves their orientations, as it preserves edge coloring,
and $\col\left(vw\right)=-\col\left(wv\right)$. However, $G$ acts
transitively on the \emph{oriented} edges of color 2, i.e.\ flipping
their orientations as well, which is possible since $2\equiv-2\mod4$.
If $E_{i}$ is the stabilizer (as a cell) of some fundamental edge
$e_{i}$ of color $i$, and $\xi:E_{2}\rightarrow\pm1$ is the character
which indicates whether an element of $E_{2}$ fixes $e_{2}$ pointwise
or flips it, then
\[
\Omega^{1}\left(\Gamma\backslash\mathcal{B}_{4}\right)\cong L^{2}\left(\Gamma\backslash G\right)^{E_{1}}\oplus L^{2}\left(\Gamma\backslash G\right)^{E_{2},\xi},
\]
where $V^{E_{2},\xi}$ denotes the $\xi$-isotypic component of a
representation $V$ (this can be thought of as a higher ``$E$-type'').)

Coming back to $\mathcal{B}_{3}$, we fix $t_{0}=\left[v_{0},\sigma v_{0},\sigma^{2}v_{0}\right]$
as a fundamental triangle. The pointwise stabilizer of $t_{0}$ is
the Iwahori subgroup
\[
I=K\cap\sigma K\sigma^{-1}\cap\sigma^{2}K\sigma^{-2}=\left\{ \left(\begin{smallmatrix}* & * & *\\
x & * & *\\
y & z & *
\end{smallmatrix}\right)\in K\,\middle|\, x,y,z\in\pi\mathcal{O}\right\} .
\]
As for edges, $G$ acts transitively on non-oriented triangles, and
preserves triangle orientation, since a flip of a triangle would imply
a flip of some edge. Thus, the stabilizer of $t_{0}$ as a cell (both
oriented and non-oriented) is
\[
T\overset{{\scriptscriptstyle def}}{=}\stab_{G}t_{0}=\left\langle \sigma\right\rangle I=I\amalg\sigma I\amalg\sigma^{2}I,
\]
and in particular $\left\langle \sigma\right\rangle $ and $I$ commute.
Again, $f\left(gt_{0}\right)=\sqrt{\mu\left(T\right)}f\left(\Gamma g\right)$
gives a linear isometry $\Omega^{2}\left(X\right)\cong L^{2}\left(\Gamma\backslash G\right)^{T}$,
and
\[
\mu\left(T\right)=\frac{\mu\left(\Gamma\backslash G\right)}{\left|\Gamma\backslash G/T\right|}=\frac{n}{\left|X^{2}\right|}=\frac{3!}{k_{0}k_{1}}=\frac{3}{\left(q^{2}+q+1\right)\left(q+1\right)}.
\]

\medskip{}

Let us denote $K_{0}=K$, $K_{1}=E$, and $K_{2}=T$, so that $\Omega^{i}\left(X\right)\cong L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$.
As $I\leq E\leq K$ and $I\leq T$, the three spaces $L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$
are contained in $L^{2}\left(\Gamma\backslash G\right)^{I}$ - the
subspace of Iwahori-fixed vectors. The Iwahori-Hecke algebra of $G$
is $\mathcal{H}=C_{c}\left(I\backslash G/I\right)$, which stands
for compactly supported, bi-$I$-invariant complex functions on $G$.
Multiplication in $\mathcal{H}$ is given by convolution, and if $\left(\rho,V\right)$
is a representation of $G$, then $\left(\overline{\rho},V^{I}\right)$
is a representation of $\mathcal{H}$, by $\overline{\rho}\left(\eta\right)v=\int_{G}\eta\left(g\right)\rho\left(g\right)v\, dg$.

We proceed to construct elements in $\mathcal{H}$ whose actions on
the subspaces $L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$ of $L^{2}\left(\Gamma\backslash G\right)$
coincide with the boundary and coboundary operators in the chain complex
of simplicial forms on $X$. For a bi-$I$-invariant set $S\subseteq G$,
let $\one_{S}$ denote the corresponding characteristic function,
as an element of $\mathcal{H}$.
\begin{prop}
\label{prop:Iwahori-boundaries}Let
\begin{equation}
\begin{aligned}\partial_{1} & =\tfrac{1}{\sqrt{\mu\left(E\right)}}\left(\one_{K\sigma^{2}}-\one_{K}\right) &  &  & \delta_{1} & =\tfrac{1}{\sqrt{\mu\left(E\right)}}\left(\one_{\sigma K}-\one_{K}\right)\\
\partial_{2} & =\tfrac{1}{\sqrt{\mu\left(E\right)\mu\left(T\right)}}\cdot\one_{ET} &  &  & \delta_{2} & =\tfrac{1}{\sqrt{\mu\left(E\right)\mu\left(T\right)}}\cdot\one_{TE}.
\end{aligned}
\label{eq:co_boundary_defs}
\end{equation}
Then each $\partial_{i}\in\mathcal{H}$ takes $L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$
to $L^{2}\left(\Gamma\backslash G\right)^{K_{i-1}}$, and acts as
the boundary operator $\partial_{i}:\Omega^{i}\left(X\right)\rightarrow\Omega^{i-1}\left(X\right)$
with respect to the identifications of $\Omega^{i}\left(X\right)$
with $L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$. The parallel
statements hold for $\delta_{i}\in\mathcal{H}$ and $\delta_{i}:\Omega^{i-1}\rightarrow\Omega^{i}$
as well. \end{prop}
\begin{proof}
As $\partial_{i}$ is constant on right $K_{i-1}$ cosets, the image
of its action on any $G$-representation $V$ is in $V^{K_{i-1}}$,
and similarly for $\delta_{i}$ and $K_{i}$ (note that $\sigma K=E\sigma K$
as $\sigma^{-1}E\sigma$ fixes $v_{0}$). Observe $V=L^{2}\left(\Gamma\backslash G\right)$
and $f\in V^{E}\cong\Omega^{1}\left(E\right)$, and denote by $\left\{ k\right\} _{kE\in\nicefrac{K}{E}}$
an arbitrary choice of a left transversal for $E$ in $K$. For any
$gv_{0}\in X^{0}$ we have
\begin{align*}
\left(\one_{K}f\right)\left(gv_{0}\right) & =\left(\one_{K}f\right)\left(\Gamma g\right)=\int_{G}\one_{K}\left(x\right)\left(xf\right)\left(\Gamma g\right)dx=\int_{K}\left(xf\right)\left(\Gamma g\right)dx\\
 & =\int_{K}f\left(\Gamma gx\right)dx=\sum_{kE\in\nicefrac{K}{E}}\int_{E}f\left(\Gamma gke\right)de=\sum_{kE\in\nicefrac{K}{E}}\int_{E}f\left(\Gamma gk\right)de\\
 & =\mu\left(E\right)\sum_{kE\in\nicefrac{K}{E}}f\left(\Gamma gk\right)=\sqrt{\mu\left(E\right)}\sum_{kE\in\nicefrac{K}{E}}f\left(gke_{0}\right),
\end{align*}
where the first and last equalities are the identifications of $L^{2}\left(\Gamma\backslash G\right)^{K_{i}}$
with $\Omega^{i}\left(X\right)$. The group $K$ acts transitively
on the $q^{2}+q+1$ edges in $\mathcal{B}$ with origin $v_{0}$ and
color $1$, which correspond to the projective points in $\nicefrac{\mathcal{O}^{3}}{\pi\cdot\mathcal{O}^{3}}\cong\mathbb{F}_{q}^{3}$.
Thus, the color-$1$ edges with origin $gv_{0}$ (for any $g\in G$)
are precisely $\left\{ gke_{0}\right\} _{kE\in\nicefrac{K}{E}}$,
giving
\begin{equation}
\frac{1}{\sqrt{\mu\left(E\right)}}\left(\one_{K}f\right)\left(gv_{0}\right)=\sum_{kE\in\nicefrac{K}{E}}f\left(gke_{0}\right)=\sum_{{\orig e=gv_{0}\atop \col e=1}}f\left(e\right)=-\sum_{{\term e=gv_{0}\atop \col e=2}}f\left(e\right).\label{eq:boundary_1_1}
\end{equation}
In a similar manner, $\sigma^{2}e_{0}=\left[\sigma^{2}v_{0},v_{0}\right]$
has color $1$ and terminates in $v_{0}$, and $K$ act transitively
on such edges (which correspond to projective lines in $\nicefrac{\mathcal{O}^{3}}{\pi\cdot\mathcal{O}^{3}}$).
Thus, the color $1$ edges which terminate in $gv_{0}$ are $\left\{ gk\sigma^{2}e_{0}\right\} _{k\in K}$.
We have $K\sigma^{2}=\left(\sigma K\right)^{-1}=\left(E\sigma K\right)^{-1}=K\sigma^{2}E$,
and if $\left\{ k\sigma^{2}\right\} _{k\sigma^{2}E\in\nicefrac{K\sigma^{2}E}{E}}$
is a left transversal of $E$ in $K\sigma^{2}E$ then
\begin{equation}
\begin{aligned}\left(\one_{K\sigma^{2}}f\right)\left(gv_{0}\right) & =\int_{K\sigma^{2}E}f\left(\Gamma gx\right)dx=\sum_{k\sigma^{2}E\in\nicefrac{K\sigma^{2}E}{E}}\int_{E}f\left(\Gamma gk\sigma^{2}e\right)de\\
 & =\sqrt{\mu\left(E\right)}\sum_{k\sigma^{2}E\in\nicefrac{K\sigma^{2}E}{E}}f\left(gk\sigma^{2}e_{0}\right)=\sqrt{\mu\left(E\right)}\sum_{{\term e=gv_{0}\atop \col e=1}}f\left(e\right).
\end{aligned}
\label{eq:boundary_1_2}
\end{equation}
Together with \prettyref{eq:boundary_1_1}, this implies that the
diagram
\[
\xymatrix{L^{2}\left(\Gamma\backslash G\right)^{K}\ar@{-}[d]|\cong &  & L^{2}\left(\Gamma\backslash G\right)^{E}\ar[ll]_{\partial_{1}\in\mathcal{H}}\ar@{-}[d]|\cong\\
\Omega^{0}\left(X\right) &  & \Omega^{1}\left(X\right)\ar[ll]_{\partial_{1}}
}
\]
commutes (which should justify the abuse of notation). The reasoning
for $\partial_{2}$ is similar, save for the fact that $T\nleq E$
(in fact, $E\cap T=I$, as a common stabilizer of $t_{0}$ and $e_{0}$
must fix $t_{0}$ pointwise.) We observe that $E$ acts transitively
on $\left(q+1\right)$ triangles containing $e_{0}$, which correspond
to incidences of projective lines and points in $\nicefrac{\mathcal{O}^{3}}{\pi\cdot\mathcal{O}^{3}}$.
Therefore, for $f\in V=L^{2}\left(\Gamma\backslash G\right)^{T}$
and $ge_{0}\in X^{1}$,
\begin{align*}
\tfrac{1}{\sqrt{\mu\left(E\right)\mu\left(T\right)}}\left(\one_{ET}f\right)\left(ge_{0}\right) & =\tfrac{1}{\sqrt{\mu\left(T\right)}}\left(\one_{ET}f\right)\left(\Gamma g\right)=\sqrt{\mu\left(T\right)}\sum_{eT\in\nicefrac{ET}{T}}f\left(\Gamma ge\right)\\
 & =\sum_{eT\in\nicefrac{ET}{T}}f\left(get_{0}\right)=\negthickspace\negthickspace\sum_{\tau\in X^{2}\,:\, e_{0}\in\partial\tau}\negthickspace\negthickspace\negthickspace\negthickspace f\left(g\tau\right)=\negthickspace\negthickspace\sum_{\tau\in X^{2}\,:\, ge_{0}\in\partial\tau}\negthickspace\negthickspace\negthickspace\negthickspace f\left(\tau\right),
\end{align*}
agreeing with $\partial_{2}:\Omega^{2}\rightarrow\Omega^{1}$.

The coboundary operators can be analyzed in a similar manner, but
they also follow from the unitary structure: $\mathcal{H}$ has a
$*$-algebra structure induced by the group inversion, namely $\eta^{*}\left(g\right)=\overline{\eta\left(g^{-1}\right)}$.
For any unitary representation $\left(\rho,V\right)$ of $G$, the
induced representation $\left(\overline{\rho},V^{I}\right)$ is unitary,
i.e.\ $\overline{\rho}\left(\eta\right)^{*}=\overline{\rho}\left(\eta^{*}\right)$
(this uses unimodularity of $G$). For $V=L^{2}\left(\Gamma\backslash G\right)$
this gives
\[
\partial_{1}^{*}=\tfrac{1}{\sqrt{\mu\left(E\right)}}\left(\overline{\one_{\left(K\sigma^{2}\right)^{-1}}}-\overline{\one_{K^{-1}}}\right)=\tfrac{1}{\sqrt{\mu\left(E\right)}}\left(\one_{\sigma K}-\one_{K}\right)
\]
and similarly for $\partial_{2}^{*}$.
\end{proof}
Since $\Gamma$ is cocompact, $L^{2}\left(\Gamma\backslash G\right)$
decomposes as a sum of irreducible unitary representations, $L^{2}\left(\Gamma\backslash G\right)=\bigoplus_{\alpha}W_{\alpha}$,
and $\Omega^{i}\left(X\right)\cong L^{2}\left(\Gamma\backslash G\right)^{K_{i}}=\bigoplus_{\alpha}W_{\alpha}^{K_{i}}\leq\bigoplus_{\alpha}W_{\alpha}^{I}$.
Each $W_{\alpha}^{I}$ is a sub-$\mathcal{H}$-representation, so
that the operators $\partial_{i},\delta_{i}$ decompose with respect
to this sum, and thus the Laplacians as well, giving $\mathrm{\Spec}\Delta_{i}^{\pm}=\bigcup_{\alpha}\Spec\Delta_{i}^{\pm}\big|_{W_{\alpha}^{K_{i}}}$,
with the correct multiplicities. To understand the spectra it is enough
look at the $W_{\alpha}$ which are \emph{Iwahori-spherical}, namely,
contain $I$-fixed vectors (for otherwise $W_{\alpha}^{K_{i}}=0$).
Furthermore, as $\Delta_{i}^{\pm}$ act by elements in $\mathcal{H}$,
the isomorphism type of $W_{\alpha}^{I}$ as an $\mathcal{H}$-module
already determines the spectrum of $\Delta_{i}^{\pm}$ on $W_{\alpha}^{K_{i}}$.
By \cite[Prop.\ 2.6]{casselman1980unramified}, every Iwahori-spherical
representation is isomorphic to a subrepresentation of a \emph{principal
series representation}. The principal series representation $V_{\mathfrak{z}}$
with \emph{Satake parameters} $\mathfrak{z}=\left(z_{1},z_{2},z_{3}\right)$,
where $z_{i}\in\mathbb{C}$ and $z_{1}z_{2}z_{3}=1$, is obtained
as follows: The standard Borel group $B=\left\{ \left(\begin{smallmatrix}* & * & *\\
0 & * & *\\
0 & 0 & *
\end{smallmatrix}\right)\in G\right\} $ admits a character $\chi_{\mathfrak{z}}\left(b\right)=\prod_{i=1}^{3}z_{i}^{\ord_{\pi}b_{ii}}$
(here $z_{1}z_{2}z_{3}=1$ ensures that $\chi_{\mathfrak{z}}$ is
well defined on $\mathrm{PGL}$), and $V_{\mathfrak{z}}$ is the unitary
induction of $\chi_{\mathfrak{z}}$ from $B$ to $G$, namely
\begin{equation}
V_{\mathfrak{z}}=\mathrm{UInd}_{B}^{G}\chi_{\mathfrak{z}}=\left\{ \left(\mathrm{smooth}\right)\: f:G\rightarrow\mathbb{C}\,\middle|\, f\left(bg\right)=\delta^{-\frac{1}{2}}\left(b\right)\chi_{\mathfrak{z}}\left(b\right)f\left(g\right)\;\forall b\in B\right\} ,\label{eq:UInd}
\end{equation}
where $\delta\left(b\right)=\left|b_{33}\right|^{2}/\left|b_{11}\right|^{2}$
is the modular character of $B$. For obvious reasons, it is convenient
to introduce the notation
\[
\widetilde{\chi}_{\mathfrak{z}}\left(b\right)=\delta^{-\frac{1}{2}}\left(b\right)\chi_{\mathfrak{z}}\left(b\right)=\frac{\left|b_{11}\right|}{\left|b_{33}\right|}\prod_{i=1}^{3}z_{i}^{\ord_{\pi}b_{ii}}=\left(\frac{z_{1}}{q}\right)^{\ord_{\pi}b_{11}}z_{2}^{\ord_{\pi}b_{22}}\left(qz_{3}\right)^{\ord_{\pi}b_{33}}.
\]

Let us stress the following point: having decomposed $L^{2}\left(\Gamma\backslash G\right)=\bigoplus_{\alpha}W_{\alpha}$,
and found some $\Delta_{i}^{\pm}$-eigenform $f\in W_{\alpha}^{K_{i}}\leq\Omega^{i}\left(X\right)$,
we can lift it to a $\Gamma$-periodic form $\tilde{f}\in{}^{\Gamma}\Omega^{i}\left(\mathcal{B}\right)$,
which is still a $\Delta_{i}^{\pm}$-eigenform (as $\Delta_{i}^{\pm}$
act in a local manner). For some Satake parameters $\mathfrak{z}$
there is an embedding of $G$-representations $\Psi:W_{\alpha}\rightarrow V_{\mathfrak{z}}$,
and naturally $\Psi f\in V_{\mathfrak{z}}^{K_{i}}$. Since the elements
in the explicit construction of $V_{\mathfrak{z}}$ in \prettyref{eq:UInd}
are complex functions on $G$, the ones in $V_{\mathfrak{z}}^{K_{i}}$
can also be seen as $i$-forms on $\mathcal{B}$. Thus, both $\tilde{f}$
and $\Psi f$ are in $\Omega^{i}\left(\mathcal{B}\right)$, and they
are $\Delta_{i}^{\pm}$-eigenforms with the same eigenvalue (as $\Psi$
is $\mathcal{H}$-equivariant). However, they are not the same, as
$\tilde{f}$ attains finitely many values and $\Psi f$ infinitely
many, in general.

While the forms $\tilde{f}$ and $\Psi f$ are different, their averages
on $K_{i}$-orbits are the same: these averages correspond to the
associated matrix coefficient of $W_{\alpha}\cong\Psi W_{\alpha}$.
When these matrix coefficient are in $L^{2+\varepsilon}\left(G\right)$
for every $\varepsilon>0$, the representation $W_{\alpha}$ is said
to be \emph{tempered}. For any irreducible unitary representation
$V$ and $v\in V^{K_{i}}$, the matrix coefficient $\phi\left(g\right)=\left\langle gv,v\right\rangle $
is bi-$K_{i}$-invariant, and if $V$ is tempered then $\phi$ is
an approximate $L^{2}-$eigenform in $\Omega^{i}\left(\mathcal{B}\right)$,
so that $\Spec\Delta_{i}^{\pm}\big|_{V^{K_{i}}}$ is contained in
the $L^{2}$-spectrum of $\Delta_{i}^{\pm}$ on $\mathcal{B}$.

\section{Analysis of the principal series}

While in general an irreducible Iwahori-spherical representation $W$
is only isomorphic to a subrepresentation of $V_{\mathfrak{z}}$,
it is simpler to consider the action of $\partial_{i},\delta_{i}$
and $\Delta_{i}^{\pm}$ on $V_{\mathfrak{z}}$, and later find which
eigenvalues are attained on $W$. By the Iwasawa decomposition $G=BK$,
for $f\in V_{\mathfrak{z}}^{K}$ and $g=bk\in G$ one has $f\left(g\right)=f\left(bk\right)=\widetilde{\chi}_{\mathfrak{z}}\left(b\right)f\left(1\right)$,
so that $V_{\mathfrak{z}}^{K}$ is at most one-dimensional. In fact,
it is one-dimensional, since $b\in B\cap K$ implies $b_{11},b_{22},b_{33}\in\mathcal{O^{\times}}$,
so that $f^{K}\left(bk\right)\overset{{\scriptscriptstyle def}}{=}\widetilde{\chi}_{\mathfrak{z}}\left(b\right)$
is well defined and spans $V_{\mathfrak{z}}^{K}$. Similarly, identifying
$S_{3}$ with the permutation matrices in $G$, the so-called Iwahori-Bruhat\emph{
}decomposition $G=\coprod_{w\in S_{3}}BwI$ shows that $\dim V_{\mathfrak{z}}^{I}=6$,
with basis $\left\{ f_{w}^{I}\right\} _{w\in S_{3}}$ defined by $f_{w}^{I}\left(w'\right)=\delta_{w,w'}$
(note that $f^{K}=\sum_{w\in S_{3}}f_{w}^{I}$). Also, $\left(1\,2\right)\in E$
(see \prettyref{eq:E-def}) implies $G=\coprod_{w\in A_{3}}BwE$,
and $\dim V_{\mathfrak{z}}^{E}=3$ with basis $\left\{ f_{w}^{E}\right\} _{w\in A_{3}}$,
where $f_{w}^{E}\overset{{\scriptscriptstyle def}}{=}f_{w}^{I}+f_{w\cdot\left(1\,2\right)}^{I}$
satisfy $f_{w}^{E}\left(w'\right)=\delta_{w,w'}$ for $w,w'\in A_{3}$.
For $T$ things are slightly more complicated, since $T\nleq K$:
here $G=BT\amalg B\left(1\,2\right)T$, and $\dim V_{\mathfrak{z}}^{T}=2$.
One can then define $f_{w}^{T}\left(w'\right)\overset{{\scriptscriptstyle def}}{=}\delta_{w,w'}$
for $w,w'\in\left\{ \left(\,\right),\left(1\,2\right)\right\} $,
and this gives
\begin{equation}
\begin{aligned}f_{\left(\,\right)}^{T} & =f_{\left(\,\right)}^{I}+\frac{1}{qz_{3}}f_{\left(3\,2\,1\right)}^{I}+\frac{z_{1}}{q}f_{\left(1\,2\,3\right)}^{I}\\
f_{\left(1\,2\right)}^{T} & =f_{\left(1\,2\right)}^{I}+z_{2}f_{\left(2\,3\right)}^{I}+\frac{1}{qz_{3}}f_{\left(1\,3\right)}^{I}\,;
\end{aligned}
\label{eq:T-basis}
\end{equation}
indeed, if $c_{w}$ is the coefficient of $f_{w}^{I}$ in $f_{\left(\,\right)}^{T}$,
then
\[
c_{\left(1\,2\,3\right)}=f_{\left(\,\right)}^{T}\left(\left(1\,2\,3\right)\right)=f_{\left(\,\right)}^{T}\left(\left(\begin{smallmatrix}\pi\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)\sigma^{2}\right)=f_{\left(\,\right)}^{T}\left(\left(\begin{smallmatrix}\pi\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)\right)=\widetilde{\chi}_{\mathfrak{z}}\left(\left(\begin{smallmatrix}\pi\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)\right)=\frac{z_{1}}{q},
\]
and the other coefficients in \prettyref{eq:T-basis} are obtained
similarly.

Now, let $\Omega_{\mathfrak{z}}^{i}\left(\mathcal{B}\right)$ be the
embedding of $V_{\mathfrak{z}}^{K_{i}}$ in $\Omega^{i}\left(\mathcal{B}\right)$
given by the explicit construction \prettyref{eq:UInd}. Any $f\in\Omega_{\mathfrak{z}}^{0}\left(\mathcal{B}\right)$
is determined by its value on $v_{0}$, namely $f=f\left(v_{0}\right)f^{K}$.
Similarly, the value on $e_{0}$, $e_{1}=\left(3\,2\,1\right)e_{0}$
and $e_{2}=\left(1\,2\,3\right)e_{0}$ determine a unique element
in $\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)$, and likewise
for $t_{0},t_{1}=\left(1\,2\right)t_{0}$ and $\Omega_{\mathfrak{z}}^{2}\left(\mathcal{B}\right)$.
As $S_{3}\leq K$, all of $v_{0},e_{0},e_{1},e_{2},t_{0},t_{1}$ are
contained in the star around $v_{0}$ in $\mathcal{B}$, and we can
understand $\partial_{i}\big|_{\Omega_{\mathfrak{z}}^{i}\left(\mathcal{B}\right)}$
and $\delta_{i}\big|_{\Omega_{\mathfrak{z}}^{i-1}\left(\mathcal{B}\right)}$
by evaluation on $\mathrm{star}\left(v_{0}\right)$ alone (see Figure
\ref{fig:v0-star}).

\begin{figure}[h]
\noindent \begin{centering}
\includegraphics{building}
\par\end{centering}

\caption{\label{fig:v0-star}The star of $v_{0}$ in $\mathcal{B}$.}
\end{figure}

Thus,
\begin{align*}
\left(\delta_{1}f^{K}\right)\left(e_{0}\right) & =f^{K}\left(\sigma v_{0}\right)-f^{K}\left(v_{0}\right)=f^{K}\left(\left(\begin{smallmatrix}1\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)v_{0}\right)-1=\widetilde{\chi}_{\mathfrak{z}}\left(\begin{smallmatrix}1\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)-1=qz_{3}-1\\
\left(\delta_{1}f^{K}\right)\left(e_{1}\right) & =f^{K}\left(\left(3\,2\,1\right)\sigma v_{0}\right)-f^{K}\left(v_{0}\right)=f^{K}\left(\left(\begin{smallmatrix}1\\
 & \pi\\
 &  & 1
\end{smallmatrix}\right)v_{0}\right)-1=z_{2}-1\\
\left(\delta_{1}f^{K}\right)\left(e_{2}\right) & =f^{K}\left(\left(1\,2\,3\right)\sigma v_{0}\right)-f^{K}\left(v_{0}\right)=f^{K}\left(\left(\begin{smallmatrix}\pi\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)v_{0}\right)-1=\frac{z_{1}}{q}-1
\end{align*}
gives
\begin{equation}
\left[\delta_{1}\big|_{\Omega_{\mathfrak{z}}^{0}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{E}}^{\negmedspace\left\{ \smash{f^{K}}\right\} }=\left(\begin{matrix}qz_{3}-1\\
z_{2}-1\\
\frac{z_{1}}{q}-1
\end{matrix}\right),\label{eq:coboundary_1_satake}
\end{equation}
where $\mathfrak{B}^{E}$ denotes the ordered basis $f_{\left(\,\right)}^{E}$,
$f_{\left(3\,2\,1\right)}^{E}$, $f_{\left(1\,2\,3\right)}^{E}$.

The edges of color $1$ with origin $v_{0}$ are $e_{0}$, $\left(\begin{smallmatrix}1\\
 & 1 & x\\
 &  & 1
\end{smallmatrix}\right)e_{1}$ for $x\in\mathbb{F}_{q}=\nicefrac{\mathcal{O}}{\pi\mathcal{O}}$,
and $\left(\begin{smallmatrix}1 & x & y\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)e_{2}$ with $x,y\in\mathbb{F}_{q}$. As $\widetilde{\chi}_{\mathfrak{z}}$
is trivial on upper-triangular unipotent matrices, every $f\in\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)$
is constant on these $q$ translations of $e_{1}$, and on the $q^{2}$
translations of $e_{2}$ (in particular this implies that $\oplus_{\mathfrak{z}}\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)$
is a proper subspace of $\Omega^{1}\left(\mathcal{B}\right)$). Thus,
\prettyref{eq:boundary_1_1} implies that for $f\in\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)$
we have $\left(\mu\left(E\right)^{-\frac{1}{2}}\one_{K}f\right)\left(v_{0}\right)=f\left(e_{0}\right)+qf\left(e_{1}\right)+q^{2}f\left(e_{2}\right)$.
The edges of color $1$ which terminate in $v_{0}$ are
\[
\left[\sigma^{2}v_{0},v_{0}\right]=\sigma^{2}e_{0}=\left(\begin{smallmatrix} &  & 1\\
\pi\\
 & \pi
\end{smallmatrix}\right)e_{0}=\left(\begin{smallmatrix}1\\
 & \pi\\
 &  & \pi
\end{smallmatrix}\right)\left(1\,2\,3\right)e_{0}=\left(\begin{smallmatrix}1\\
 & \pi\\
 &  & \pi
\end{smallmatrix}\right)e_{2},
\]
and similarly $\left(\begin{smallmatrix}\pi & x\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)e_{1}$ and $\left(\begin{smallmatrix}\pi &  & x\\
 & \pi & y\\
 &  & 1
\end{smallmatrix}\right)e_{0}$ ($x,y\in\mathbb{F}_{q}$). By \prettyref{eq:boundary_1_2},
\begin{align*}
\left(\tfrac{1}{\sqrt{\mu\left(E\right)}}\one_{K\sigma^{2}E}f\right)\left(v_{0}\right) & =f\left(\left(\begin{smallmatrix}1\\
 & \pi\\
 &  & \pi
\end{smallmatrix}\right)e_{2}\right)+\sum_{x\in\mathbb{F}_{q}}f\left(\left(\begin{smallmatrix}\pi & x\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)e_{1}\right)+\sum_{x,y\in\mathbb{F}_{q}}f\left(\left(\begin{smallmatrix}\pi &  & x\\
 & \pi & y\\
 &  & 1
\end{smallmatrix}\right)e_{0}\right)\\
 & =z_{2}qz_{3}f\left(e_{2}\right)+q\cdot z_{1}z_{3}f\left(e_{1}\right)+q^{2}\cdot\frac{z_{1}}{q}z_{2}f\left(e_{0}\right),
\end{align*}
and in total (see \prettyref{eq:co_boundary_defs})
\begin{equation}
\left[\partial_{1}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)}\right]_{\negmedspace\left\{ \smash{f^{K}}\right\} }^{\mathfrak{B}^{E}}=\left(\begin{matrix}\frac{q}{z_{3}}-1 & \frac{q}{z_{2}}-q & \frac{q}{z_{1}}-q^{2}\end{matrix}\right).\label{eq:boundary_1_satake}
\end{equation}
As $\Delta_{0}^{+}=\partial_{1}\delta_{1}$ and $\Delta_{1}^{-}=\delta_{1}\partial_{1}$,
we can now compute explicitly their action on the $\mathfrak{z}$-principal
series. Denoting $\widetilde{\mathfrak{z}}=\sum_{i=1}^{3}\left(z_{i}+z_{i}^{-1}\right)$,
we have by \prettyref{eq:coboundary_1_satake} and \prettyref{eq:boundary_1_satake}
\begin{align}
\Delta_{0}^{+}\big|_{\Omega_{\mathfrak{z}}^{0}\left(\mathcal{B}\right)} & =\left(\lambda^{K}\right)\overset{{\scriptscriptstyle def}}{=}\left(k_{0}-q\widetilde{\mathfrak{z}}\right),\label{eq:principle-series-e.v.-vertices}\\
\left[\Delta_{1}^{-}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{E}} & =\left(\begin{smallmatrix}{q}^{2}-qz_{{3}}-{\frac{q}{z_{{3}}}}+1 & -{q}^{2}z_{{3}}+{\frac{{q}^{2}z_{{3}}}{z_{{2}}}}+q-{\frac{q}{z_{{2}}}} & -{q}^{3}z_{{3}}+{\frac{{q}^{2}z_{{3}}}{z_{{1}}}}+{q}^{2}-{\frac{q}{z_{{1}}}}\\
\noalign{\smallskip}{\frac{qz_{{2}}}{z_{{3}}}}-z_{{2}}-{\frac{q}{z_{{3}}}}+1 & -qz_{{2}}+2\, q-{\frac{q}{z_{{2}}}} & -{q}^{2}z_{{2}}+{q}^{2}+{\frac{qz_{{2}}}{z_{{1}}}}-{\frac{q}{z_{{1}}}}\\
\noalign{\smallskip}-{\frac{q}{z_{{3}}}}-{\frac{z_{{1}}}{q}}+{\frac{z_{{1}}}{z_{{3}}}}+1 & q-z_{{1}}-{\frac{q}{z_{{2}}}}+{\frac{z_{{1}}}{z_{{2}}}} & {q}^{2}-qz_{{1}}-{\frac{q}{z_{{1}}}}+1
\end{smallmatrix}\right).\nonumber
\end{align}
Note that $\Delta_{0}^{+}$ agrees with the computation of the spectrum
of the Hecke operators in \cite{macdonald1979symmetric,li2004ramanujan,Lubotzky2005a},
as $\Delta_{0}^{+}=k_{0}\cdot I-\sum_{i=1}^{d-1}A_{i}$ (here $A_{i}$
is the $i$-th Hecke operator on $\mathcal{B}_{d}$, loc.\ cit.).
We also understand $\Delta_{1}^{-}$ well enough, as it has the eigenvalue
$\lambda^{K}$ corresponding to $\delta_{1}\Omega_{\mathfrak{z}}^{0}\left(\mathcal{B}\right)$
(since $\Delta_{1}^{-}\delta_{1}f^{K}=\delta_{1}\Delta_{0}^{+}f^{K}=\lambda^{K}\delta_{1}f^{K}$),
and two zeros (which come from $\partial_{2}\Omega_{\mathfrak{z}}^{2}\left(\mathcal{B}\right)$).
However, we can now proceed to compute as easily the edge/triangle
spectrum: for $f\in\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)$,
\begin{align*}
\left(\delta_{2}f\right)\left(t_{0}\right) & =\sum\nolimits _{i=0}^{2}f\left(\sigma^{i}e_{0}\right)=f\left(e_{0}\right)+f\left(\left(\begin{smallmatrix}1\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)\left(3\,2\,1\right)e_{0}\right)+f\left(\left(\begin{smallmatrix}1\\
 & \pi\\
 &  & \pi
\end{smallmatrix}\right)\left(1\,2\,3\right)e_{0}\right)\\
 & =f\left(e_{0}\right)+qz_{3}f\left(e_{1}\right)+z_{2}qz_{3}f\left(e_{2}\right)\\
\left(\delta_{2}f\right)\left(t_{1}\right) & =\sum\nolimits _{i=0}^{2}f\left(\left(1\,2\right)\sigma^{i}e_{0}\right)=f\left(e_{0}\right)+f\left(\left(\begin{smallmatrix}1\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)\left(1\,2\,3\right)e_{0}\right)+f\left(\left(\begin{smallmatrix}\pi\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)\left(3\,2\,1\right)e_{0}\right)\\
 & =f\left(e_{0}\right)+qz_{3}f\left(e_{2}\right)+z_{1}z_{3}f\left(e_{1}\right)
\end{align*}
show that $\left[\delta_{2}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{T}}^{\mathfrak{B}^{E}}=\left(\begin{matrix}1 & qz_{3} & qz_{2}z_{3}\\
1 & z_{1}z_{3} & qz_{3}
\end{matrix}\right)$, where $\mathfrak{B}^{T}$ is the ordered basis $f_{\left(\,\right)}^{T},f_{\left(1\,2\right)}^{T}$
(and at this point one can verify directly that $\delta_{2}\delta_{1}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)}=0$).

The triangles containing $e_{0}$ are obtained by adjoining $\sigma^{2}v_{0}$
(which gives $t_{0}$) and $\left(\begin{smallmatrix}\pi & x\\
 & 1\\
 &  & \pi
\end{smallmatrix}\right)v_{0}$ ($x\in\mathbb{F}_{q}$), giving $\left(\begin{smallmatrix}1 & x\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)t_{1}$. This yields $\left(\partial_{2}f\right)\left(e_{0}\right)=f\left(t_{0}\right)+qf\left(t_{1}\right)$,
but for $e_{1},e_{2}$ we need to work a little harder, and use \prettyref{eq:T-basis}:
\begin{align*}
\left(\partial_{2}f\right)\left(e_{1}\right) & =\left(\partial_{2}f\right)\left(\left(3\,2\,1\right)e_{0}\right)=f\left(\left(3\,2\,1\right)t_{0}\right)+\sum_{x\in\mathbb{F}_{q}}f\left(\left(3\,2\,1\right)\left(\begin{smallmatrix}1 & x\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)t_{1}\right)\\
 & =\frac{1}{qz_{3}}f\left(t_{0}\right)+f\left(\left(3\,2\,1\right)t_{1}\right)+\sum_{x\in\mathbb{F}_{q}^{\times}}f\left(\left(3\,2\,1\right)\left(\begin{smallmatrix}1 & x\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)\left(1\,2\right)t_{0}\right)\\
 & =\frac{1}{qz_{3}}f\left(t_{0}\right)+f\left(\left(2\,3\right)t_{0}\right)+\sum_{x\in\mathbb{F}_{q}^{\times}}f\left(\left(\begin{smallmatrix}-\frac{1}{x} &  & 1\\
 & 1\\
 &  & x
\end{smallmatrix}\right)\left(3\,2\,1\right)\left(\begin{smallmatrix}1 & \frac{1}{x}\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)t_{0}\right)\\
 & =\frac{1}{qz_{3}}f\left(t_{0}\right)+z_{2}f\left(t_{1}\right)+\sum_{x\in\mathbb{F}_{q}^{\times}}f\left(\left(3\,2\,1\right)t_{0}\right)=\frac{1}{z_{3}}f\left(t_{0}\right)+z_{2}f\left(t_{1}\right)
\end{align*}
\begin{align*}
\left(\partial_{2}f\right)\left(e_{2}\right) & =\left(\partial_{2}f\right)\left(\left(1\,2\,3\right)e_{0}\right)=f\left(\left(1\,2\,3\right)t_{0}\right)+\sum_{x\in\mathbb{F}_{q}}f\left(\left(1\,2\,3\right)\left(\begin{smallmatrix}1 & x\\
 & 1\\
 &  & 1
\end{smallmatrix}\right)t_{1}\right)\\
 & =\frac{z_{1}}{q}f\left(t_{0}\right)+\sum_{x\in\mathbb{F}_{q}}f\left(\left(\begin{smallmatrix}1\\
 & 1 & x\\
 &  & 1
\end{smallmatrix}\right)\left(1\,3\right)t_{0}\right)=\frac{z_{1}}{q}f\left(t_{0}\right)+\frac{1}{z_{3}}f\left(t_{1}\right),
\end{align*}
 so that $\left[\partial_{2}\big|_{\Omega_{\mathfrak{z}}^{2}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{E}}^{\mathfrak{B}^{T}}=\left(\begin{matrix}1 & q\\
\nicefrac{1}{z_{3}} & z_{2}\\
\nicefrac{z_{1}}{q} & \nicefrac{1}{z_{3}}
\end{matrix}\right)$, giving
\begin{align*}
\left[\Delta_{1}^{+}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{E}} & =\left(\begin{array}{ccc}
q+1 & \frac{q}{z_{2}}+qz_{{3}} & {q}^{2}z_{{3}}+\frac{q}{z_{1}}\\
\noalign{\smallskip}{z_{{2}}+\frac{1}{z_{{3}}}} & q+1 & \frac{q}{z_{1}}+qz_{{2}}\\
\noalign{\smallskip}{\frac{z_{1}}{q}+\frac{1}{z_{3}}} & \frac{1}{z_{2}}+z_{{1}} & q+1
\end{array}\right)\\
\left[\Delta_{2}^{-}\big|_{\Omega_{\mathfrak{z}}^{2}\left(\mathcal{B}\right)}\right]_{\mathfrak{B}^{T}} & =\left(\begin{array}{cc}
q+2 & \frac{q}{z_{1}}+qz_{{2}}+q\\
\noalign{\smallskip}\frac{1}{z_{2}}+z_{{1}}+1 & 2q+1
\end{array}\right).
\end{align*}
Recalling that $\lambda^{K}=k_{0}-q\widetilde{\mathfrak{z}}=2\left(q^{2}+q+1\right)-q\left(\sum z_{i}+\frac{1}{z^{i}}\right)$,
one obtains
\begin{equation}
\begin{aligned}\Spec\Delta_{1}^{+}\big|_{\Omega_{\mathfrak{z}}^{1}\left(\mathcal{B}\right)} & =\left\{ \lambda_{0}^{E},\lambda_{\pm}^{E}\right\} \overset{{\scriptscriptstyle def}}{=}\left\{ 0,\tfrac{3}{2}\left(q+1\right)\pm\tfrac{1}{2}\sqrt{\left(q+1\right)^{2}+4q\left(2+\widetilde{\mathfrak{z}}\right)}\right\} \\
 & =\left\{ 0,\tfrac{3k_{1}}{2}\pm\sqrt{\left(\tfrac{3k_{1}}{2}\right)^{2}-\lambda^{K}}\right\} ,
\end{aligned}
\label{eq:principle-series-e.v.-edges}
\end{equation}
and again $\Spec\Delta_{2}^{-}\big|_{\Omega_{\mathfrak{z}}^{2}\left(\mathcal{B}\right)}=\left\{ \lambda_{\pm}^{E}\right\} $
as we have argued for $\Delta_{1}^{-}$. For $\Delta_{1}^{+}$, $\lambda_{0}^{E}=0$
is obtained on $\delta_{1}f^{K}$ (whose $f_{w}^{E}$ coefficients
were computed in \prettyref{eq:coboundary_1_satake}), and $\lambda_{\pm}^{E}$
are obtained on
\begin{align*}
f_{\pm}^{E} & =\left(\begin{array}{c}
2\left(z_{2}^{-2}+{\frac{z_{{1}}}{z_{{2}}}}\right){q}^{2}-2\left(z_{{3}}+1\right)q\\
\noalign{\smallskip}1-{q}^{2}z_{{1}}+q\left(z_{{1}}+\frac{2}{z_{2}}-\frac{2}{z_{3}}-1\right)\pm\left(qz_{{1}}-1\right)\sqrt{9k_{1}^{2}-4\lambda^{K}}\\
\noalign{\smallskip}qz_{1}\left(z_{2}^{-1}+2{z_{{1}}}+1\right)-{\frac{z_{{1}}}{z_{{2}}}}-z_{{1}}-\frac{2}{z_{2}}\pm\left(-z_{{1}}+{\frac{z_{{1}}}{z_{{2}}}}\right)\sqrt{9k_{1}^{2}-4\lambda^{K}}
\end{array}\right)^{T}\cdot\left(\begin{matrix}f_{\left(\,\right)}^{E}\\
\noalign{\medskip}f_{\left(3\,2\,1\right)}^{E}\\
\noalign{\medskip}f_{\left(1\,2\,3\right)}^{E}
\end{matrix}\right)\\
 & =2q\left(1+z_{2}+\frac{1}{z_{1}}\right)\partial_{2}f_{\left(\,\right)}^{T}+\left(q-1\pm\sqrt{9k_{1}^{2}-4\lambda^{K}}\right)\partial_{2}f_{\left(1\,2\right)}^{T}.
\end{align*}

\section{\label{sec:Unitary-Iwahori-spherical-repres}Unitary Iwahori-spherical
representations}

In general, an irreducible Iwahori-spherical representation is only
a subrepresentation of $V_{\mathfrak{z}}$. Let us denote by $W_{\mathfrak{z}}$
this subrepresentation (there is only one such). Tadic \cite{tadic1986classification}
has classified the Satake parameters for which the representation
$W_{\mathfrak{z}}$ admits a unitary structure. In \cite{kang2010zeta}
the possible $\mathfrak{z}$ for $\mathrm{PGL}_{3}\left(F\right)$
are listed, and a basis for $W_{\mathfrak{z}}\leq V_{\mathfrak{z}}$
is computed explicitly, using results from \cite{borel1976admissible,zelevinsky1980induced}.
It turns out that a unitary $E$-spherical $W_{\mathfrak{z}}$ is
of one of the following types:
\begin{enumerate}[label=(\alph{enumi})]
\item \label{enu:prin-unit}$\left|z_{i}\right|=1$ for $i=1,2,3$. In
this case $V_{\mathfrak{z}}$ is irreducible, hence $W_{\mathfrak{z}}=V_{\mathfrak{z}}$,
and $\widetilde{\mathfrak{z}}\in\left[-3,6\right]$ gives $\lambda^{K}\in\left[k_{0}-6q,k_{0}+3q\right]$
and $\lambda_{\pm}^{E}\in\mathcal{I}_{\pm}$ (see \prettyref{eq:principle-series-e.v.-edges}
and \prettyref{eq:good-bad-interval}).
\item \label{enu:bad-prin}$\mathfrak{z}=\left(c^{-2},cq^{a},cq^{-a}\right)$
for some $\left|c\right|=1$ and $0<a<\frac{1}{2}$. Here too $V_{\mathfrak{z}}$
is irreducible.
\item \label{enu:Li-e}$\mathfrak{z}=\left(\frac{c}{\sqrt{q}},c\sqrt{q},c^{-2}\right)$
for some $\left|c\right|=1$. In this case $W_{\mathfrak{z}}^{E}$
is one-dimensional, and spanned by $f_{-}^{E}$, which is proportional
to $qf_{\left(3\,2\,1\right)}^{E}-f_{\left(1\,2\,3\right)}^{E}$.
It corresponds to
\begin{align*}
\lambda_{-}^{E} & =\frac{1}{2}\left(3k_{1}-\sqrt{k_{1}^{2}+8q+4q\left(\frac{c}{\sqrt{q}}+\overline{c}\sqrt{q}+c\sqrt{q}+\frac{\overline{c}}{q}+c^{-2}+c^{2}\right)}\right)\\
 & =\frac{1}{2}\left(3k_{1}-\sqrt{q^{2}+8q\sqrt{q}\Re\left(c\right)+2q+16q\Re\left(c\right)^{2}+1+8\sqrt{q}\Re\left(c\right)}\right)\\
 & =\frac{1}{2}\left(3k_{1}-\left(q+4\sqrt{q}\Re\left(c\right)+1\right)\right)=k_{1}-2\sqrt{q}\Re\left(c\right)
\end{align*}
which lies in $\mathcal{I}$ (see \prettyref{eq:excellent-interval}).
As $f_{-}^{E}$ is not $K$-fixed, $W_{\mathfrak{z}}^{K}=0$.
\item \label{enu:Li-d}$\mathfrak{z}=\left(c\sqrt{q},\frac{c}{\sqrt{q}},c^{-2}\right)$
for some $\left|c\right|=1$. Here $W_{\mathfrak{z}}^{E}=\left\langle f_{0}^{E},f_{+}^{E}\right\rangle $,
where $f_{+}^{E}$ is proportional to $\left(q+1\right)f_{\left(\,\right)}^{E}+\left(c^{2}+\frac{c}{\sqrt{q}}\right)\left(f_{\left(3\,2\,1\right)}^{E}+f_{\left(1\,2\,3\right)}^{E}\right)$,
and $\lambda_{+}^{E}=2k_{1}+2\sqrt{q}\Re\left(c\right)$ similarly
to the computation in type \prettyref{enu:Li-e}. This time $f^{K}$
is in $W_{\mathfrak{z}}^{E}$, and corresponds to $\lambda^{K}=k_{0}-2q\Re\left(\frac{\left(q+1\right)}{\sqrt{q}}c+c^{2}\right)$.
\item \label{enu:triv}$\mathfrak{z}=\left(q,1,{\scriptscriptstyle \frac{1}{q}}\right)$.
In this case $W_{\mathfrak{z}}$ is the trivial representation $\rho:G\rightarrow\mathbb{C}^{\times}$,
and $W_{\mathfrak{z}}^{E}=W_{\mathfrak{z}}^{K}$ are spanned by $f^{K}=f_{+}^{E}$.
Since $f^{K}$ is constant and $f_{+}^{E}$ is a disorientation we
have $\lambda^{K}=0$ and $\lambda_{+}^{E}=3k_{1}$ (alternatively,
use \prettyref{eq:principle-series-e.v.-vertices} and \prettyref{eq:principle-series-e.v.-edges}).
\item \label{enu:color}$\mathfrak{z}=\left(\omega q,\omega,{\scriptscriptstyle \frac{\omega}{q}}\right)$
where $\omega=e^{\frac{2\pi i}{3}}$ or $\omega=e^{-\frac{2\pi i}{3}}$.
Here $W_{\mathfrak{z}}$ is the one-dimensional representation $\rho\left(g\right)=\omega^{\col\left(g\right)}$,
and $W_{\mathfrak{z}}^{K}=W_{\mathfrak{z}}^{E}=\left\langle f^{K}\right\rangle =\left\langle f_{0}^{E}\right\rangle $,
giving $\lambda^{K}=\frac{3k_{0}}{2}$.
\end{enumerate}
Apart from these there is the Steinberg $\mathrm{\left(Stn\right)}$
representation $\mathfrak{z}=\left({\scriptscriptstyle \frac{1}{q}},1,q\right)$.
It is not $E$-spherical, and $W^{T}$ is spanned by $f_{0}^{T}=qf_{\left(\,\right)}^{T}-f_{\left(1\,2\right)}^{T}$,
which is always in $\ker\partial_{2}=\ker\Delta_{2}^{-}$. (In \cite{kang2010zeta}
the twisted Steinberg representations $\mathfrak{z}=\left({\scriptscriptstyle \frac{\omega}{q}},\omega,\omega q\right)$
are considered as well, but these do not contribute to the simplicial
theory as they have no $K$, $E$ or $T$-fixed vectors.)

Let $X=\Gamma\backslash\mathcal{B}$ be a non-tripartite Ramanujan
complex with $L^{2}\left(\Gamma\backslash G\right)\cong\bigoplus_{i}W_{\mathfrak{z_{i}}}$,
and denote by $N_{\left(t\right)}$ the number of $W_{\mathfrak{z}_{i}}$
of type $\left(t\right)$. These are computed in \cite{kang2010zeta}
for the tripartite case, and our arguments are very similar. By the
Ramanujan assumption every Iwahori-spherical $W_{\mathfrak{z}_{i}}$
is either tempered, which are the types \prettyref{enu:prin-unit},
\prettyref{enu:Li-e}, and $\mathrm{\left(Stn\right)}$, or finite-dimensional
(types \prettyref{enu:triv}, \prettyref{enu:color}), so that $N_{\ref{enu:bad-prin}}=N_{\ref{enu:Li-d}}=0$.
The trivial representation \prettyref{enu:triv} always appears once
in $L^{2}\left(\Gamma\backslash G\right)$ as the constant functions,
so that $N_{\ref{enu:triv}}=1$.
\footnote{This gives an algebraic explanation to the fact that $3k_{1}$ always
appear in $\Spec\Delta_{1}^{+}$, unlike the graph case, where $2k_{0}\in\Spec\Delta_{0}^{+}$
only for bipartite quotients of $\mathcal{B}_{2}$.
} The color representations (type \prettyref{enu:color}) correspond
to $f\in L^{2}\left(\Gamma\backslash G\right)$ satisfying $f\left(\Gamma g\right)=\left(gf\right)\left(\Gamma\right)=\omega^{\col g}f\left(\Gamma\right)$,
which is unique up to a scalar, and well defined iff $\col\Gamma\equiv0$.
As we assume $X$ to be non-tripartite, $N_{\ref{enu:color}}=0$ (and
otherwise we would have had $N_{\ref{enu:color}}=2$). Next,
\begin{align*}
n & =\dim\Omega^{0}\left(X\right)=\sum\nolimits _{i}\dim W_{\mathfrak{z}_{i}}^{K}=N_{\ref{enu:prin-unit}}+N_{\ref{enu:triv}}+N_{\ref{enu:color}}\\
\frac{nk_{0}}{2} & =\dim\Omega^{1}\left(X\right)=\sum\nolimits _{i}\dim W_{\mathfrak{z}_{i}}^{E}=3N_{\ref{enu:prin-unit}}+N_{\ref{enu:Li-e}}+N_{\ref{enu:triv}}+N_{\ref{enu:color}}
\end{align*}
give $N_{\ref{enu:prin-unit}}=n-1$ and $N_{\ref{enu:Li-e}}=n\left(q^{2}+q-2\right)+2$.
This is summarized in Table \ref{tab:rep-info}, together with the
tripartite case, and this also completes the proof of \prettyref{thm:ram-triangle-spec}.

For completeness, we list in Table \ref{tab:rep-info} what is $W^{T}$
for each type. This gives $\frac{nk_{0}k_{1}}{6}=\dim\Omega^{2}\left(X\right)=2N_{\ref{enu:prin-unit}}+N_{\ref{enu:Li-e}}+N_{\ref{enu:triv}}+N_{\left(Stn\right)}$,
from which it follows that $N_{\left(Stn\right)}=\sum_{i=-1}^{2}\left(-1\right)^{i}\left|X^{i}\right|=\widetilde{\chi}\left(X\right)$,
the reduced Euler characteristic of $X$ (as is shown in \cite{kang2010zeta}
for the tripartite case).

\begin{table}[h]
\noindent \begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Type &  & $W^{K}$ & $\Delta_{0}^{+}$ e.v. & $W^{E}$ & $\Delta_{1}^{+}$ e.v. & $W^{T}$ & ${\mathrm{mult.}\atop \col\Gamma=0}$ & ${\mathrm{mult.}\atop \col\Gamma\neq0}$\tabularnewline
\hline
\hline
\prettyref{enu:prin-unit} & {\small tempered} & $f^{K}$ & $k_{0}+q\widetilde{\mathfrak{z}}$ & $f_{0}^{E},f_{\pm}^{E}$ & $0,\tfrac{3k_{1}}{2}\!\pm\!\sqrt{\tfrac{9k_{1}^{2}}{4}-\lambda^{K}}$ & $\delta_{1}f_{\pm}^{E}$ & $n-3$ & $n-1$\tabularnewline
\hline
\prettyref{enu:bad-prin} &  & $f^{K}$ & $k_{0}+q\widetilde{\mathfrak{z}}$ & $f_{0}^{E},f_{\pm}^{E}$ & $0,\tfrac{3k_{1}}{2}\!\pm\!\sqrt{\tfrac{9k_{1}^{2}}{4}-\lambda^{K}}$ & $\delta_{1}f_{\pm}^{E}$ & 0 & 0\tabularnewline
\hline
\prettyref{enu:Li-e} & {\small tempered} & $0$ & - & $f_{-}^{E}$ & $k_{1}-2\sqrt{q}\Re\left(c\right)$ & $\delta_{1}f_{-}^{E}$ & ${nq^{2}+nq\atop -2n+6}$ & ${nq^{2}+nq\atop -2n+2}$\tabularnewline
\hline
\prettyref{enu:Li-d} &  & $f^{K}$ & $k_{0}+q\widetilde{\mathfrak{z}}$ & $f_{0}^{E},f_{+}^{E}$ & $0,2k_{1}+2\sqrt{q}\Re\left(c\right)$ & $\delta_{1}f_{+}^{E}$ & 0 & 0\tabularnewline
\hline
\prettyref{enu:triv} & {\small trivial} & $f^{K}$ & 0 & $f_{+}^{E}$ & $3k_{1}$ & $\delta_{1}f_{+}^{E}$ & 1 & 1\tabularnewline
\hline
\prettyref{enu:color} & {\small fin.\ dim.} & $f^{K}$ & $\frac{3k_{0}}{2}$ & $f_{0}^{E}$ & 0 & 0 & 2 & 0\tabularnewline
\hline
$\mathrm{\left(Stn\right)}$ & {\small tempered} & 0 & - & 0 & - & $f_{0}^{T}$ & $\widetilde{\chi}\left(X\right)$ & $\widetilde{\chi}\left(X\right)$\tabularnewline
\hline
\end{tabular}
\par\end{centering}

\caption{\label{tab:rep-info}The representations appearing in $L^{2}\left(\Gamma\backslash G\right)$,
with the corresponding Laplacian eigenvalues, and the multiplicity
of appearance in the tripartite and non-tripartite Ramanujan cases.}
\end{table}

\section{\label{sec:comb-expansion}Combinatorial expansion}

We continue focusing on the non-tripartite Ramanujan case. We now
know that the nontrivial vertex spectrum ($\Spec\Delta_{0}^{+}\big|_{Z_{0}}$)
is highly concentrated, lying in a $k_{0}\pm O\left(\sqrt{k_{0}}\right)$
strip. The nontrivial edge spectrum is ``almost concentrated'':
there are $\approx nq^{2}$ eigenvalues in a $k_{1}\pm O\left(\sqrt{k_{1}}\right)$
strip, but also $n-1$ eigenvalues at $2k_{1}\pm O\left(1\right)$
(and the eigenvalue $3k_{1}$). Nevertheless, having a concentrated
vertex spectrum, and edge spectrum bounded away from zero is enough
to prove the Cheeger-type inequality stated in \prettyref{thm:Cheeger}.
It can be rephrased in terms of a Cheeger-type constant: for a fixed
constant $0\leq\vartheta\leq\frac{1}{3}$, define
\begin{equation}
h_{\vartheta}\left(X\right)=\min\left\{ \frac{\left|T\left(A,B,C\right)\right|n^{2}}{\left|A\right|\left|B\right|\left|C\right|}\,\middle|\,{V=A\amalg B\amalg C,\atop \left|A\right|,\left|B\right|,\left|C\right|\geq\vartheta n}\right\} \label{eq:cheeger-triangle-def}
\end{equation}
\prettyref{thm:Cheeger} then implies that
\begin{equation}
h_{\vartheta}\left(X\right)\geq\lambda_{1}\left(k_{0}-\mu_{0}\left(1+\frac{10}{9\vartheta^{3}}\right)\right).\label{eq:cheeger-ineq-triangle}
\end{equation}
For Ramanujan triangle complexes, \prettyref{thm:ram-triangle-spec}
gives $\mu_{0}=6q$ and $\lambda_{1}=q+1-2\sqrt{q}$, and therefore
we have the following:
\begin{cor}
Fix $\vartheta>0$, and let $X$ be a non-tripartite Ramanujan triangle
complex with $n$ vertices, vertex degree $k_{0}=2\left(q^{2}+q+1\right)$
and edge degree $k_{1}=q+1$. For any partition of the vertices of
$X$ into sets $A,B,C$ of sizes at least $\vartheta n,$
\[
\frac{\left|T\left(A,B,C\right)\right|n^{2}}{\left|A\right|\left|B\right|\left|C\right|}\geq\left(q+1-2\sqrt{q}\right)\left(2q^{2}+2q+2-6q\left(1+\frac{10}{9\vartheta^{3}}\right)\right).
\]
In particular,
\begin{equation}
h_{\vartheta}\left(X\right)\geq2q^{3}-O_{\vartheta}\left(q^{2.5}\right).\label{eq:Ram-Cheeger}
\end{equation}

\end{cor}
We make a few remarks:
\begin{enumerate}
\item The inequality \prettyref{eq:Ram-Cheeger} corresponds to the pseudo-random
expectation: $X$ has $\frac{1}{3!}$$nk_{0}k_{1}$ triangles, so
its triangle density is indeed ${\displaystyle \frac{\frac{1}{3}n\left(q^{2}+q+1\right)\left(q+1\right)}{{n \choose 3}}\approx\frac{2q^{3}}{n^{2}}}$.
\item If a triangle complex $X$ has a complete skeleton, then $\Spec\Delta_{0}^{+}\big|_{Z_{0}}=\left\{ n\right\} $,
so that $k_{0}=n$ and $\mu_{0}=0$. Therefore, \prettyref{thm:Cheeger}
reads $\frac{\left|T\left(A,B,C\right)\right|n^{2}}{\left|A\right|\left|B\right|\left|C\right|}\geq\lambda_{1}\cdot n$,
which is precisely the Cheeger inequality which appears in \cite{parzanchevski2012isoperimetric}
for complexes with a complete skeleton.
\item \label{enu:sublinear-sets}The reason to consider only partitions
into ``large blocks'' in \prettyref{eq:cheeger-triangle-def} is
the following: If $f\left(n\right)$ is any sub-linear function, one
can take $A\subseteq X^{0}$ to be any set of size $f\left(n\right)$,
$B$ to be $\partial A=\left\{ v\,\middle|\,\mathrm{dist}\left(v,A\right)=1\right\} $
(if $\left|B\right|<f\left(n\right)$ enlarge it by adding any vertices),
and $C$ the rest of the vertices. Assuming $n$ is large enough one
has $\left|A\right|,\left|B\right|,\left|C\right|\geq f\left(n\right)$,
and $T\left(A,B,C\right)=\varnothing$ since all triangles with a
vertex in $A$ have their other vertices in either $A$ or $B$. Thus,
we are led to partitions into blocks of linear size.
\item Another Cheeger constant for complexes with non-complete skeleton
was suggested in \cite{parzanchevski2012isoperimetric}, and related
to the spectrum in \cite{Gundert2014}:
\[
\widetilde{h}\left(X\right)=\min\limits _{V=A\coprod B\coprod C}\frac{\left|T\left(A,B,C\right)\right|\cdot n}{\left|T^{\partial}\left(A,B,C\right)\right|},
\]
where $T^{\partial}\left(A,B,C\right)$ is the set of triangle-boundaries
with one vertex in each of $A$, $B$ and $C$. This constant, however,
is not interesting for Ramanujan complexes - these are clique complexes,
so that $\widetilde{h}\left(X\right)=n$ trivially.
\end{enumerate}
We now proceed to the proof:
\begin{proof}[Proof of \prettyref{thm:Cheeger}]
Denote $\left|A\right|,\left|B\right|,\left|C\right|$ by $a,b,c$,
respectively, and define $f\in\Omega^{1}$ by
\[
f\left(vw\right)=\left\{ \begin{aligned}c &  &  & v\in A,w\in B &  &  & -c &  &  & w\in A,v\in B\\
a &  &  & v\in B,w\in C &  &  & -a &  &  & w\in B,v\in C\\
b &  &  & v\in C,w\in A &  &  & -b &  &  & w\in C,v\in A\\
0 &  &  & otherwise.
\end{aligned}
\right.
\]
Let $f_{B}=\mathbb{P}_{B^{1}}f$ and $f_{Z}=\mathbb{P}_{Z_{1}}f$.
Then
\begin{align*}
\left|T\left(A,B,C\right)\right|n^{2} & =\sum_{t\in T}\left(\delta f\right)^{2}\left(t\right)=\left\Vert \delta f\right\Vert ^{2}=\left\Vert \delta f_{Z}\right\Vert ^{2}\\
 & =\left\langle \Delta^{+}f_{Z},f_{Z}\right\rangle \geq\lambda_{1}\left\Vert f_{Z}\right\Vert ^{2}=\lambda_{1}\left(\left\Vert f\right\Vert ^{2}-\left\Vert f_{B}\right\Vert ^{2}\right).
\end{align*}
Let us denote
\begin{equation}
\mathfrak{D}=k_{0}\mathbb{P}_{B^{1}}-\Delta_{1}^{-}.\label{eq:coboundary-to-lowerlap}
\end{equation}
Any linear maps $T:V\rightarrow W$ and $S:W\rightarrow V$ satisfy
$\Spec TS\backslash\left\{ 0\right\} =\Spec ST\backslash\left\{ 0\right\} $,
and thus
\begin{align*}
\Spec\Delta_{1}^{-}\big|_{B^{1}} & =\Spec\Delta_{1}^{-}\backslash\left\{ 0\right\} =\Spec\Delta_{0}^{+}\backslash\left\{ 0\right\} =\Spec\Delta_{0}^{+}\big|_{B_{0}}\\
 & \subseteq\Spec\Delta_{0}^{+}\big|_{Z_{0}}\subseteq\left[k_{0}-\mu_{0},k_{0}+\mu_{0}\right].
\end{align*}
Together with $\Delta_{1}^{-}\big|_{Z_{1}}=0$ this implies $\left\Vert \mathfrak{D}\right\Vert \leq\mu_{0}$,
so that
\[
\left\Vert f_{B}\right\Vert ^{2}=\left\langle \mathbb{P}_{B^{1}}f,\mathbb{P}_{B^{1}}f\right\rangle =\left\langle \mathbb{P}_{B^{1}}f,f\right\rangle \leq\frac{\left|\left\langle \mathfrak{D}f,f\right\rangle \right|+\left|\left\langle \Delta_{1}^{-}f,f\right\rangle \right|}{k_{0}}\leq\frac{\mu_{0}\left\Vert f\right\Vert ^{2}+\left\Vert \partial f\right\Vert ^{2}}{k_{0}}.
\]
Using the expander mixing lemma
\footnote{See e.g.\ \cite{Parzanchevski2013} for a statement which does not
assume regularity.
} for $E\left(A,B\right)$, $E\left(B,C\right)$ and $E\left(C,A\right)$
we have
\begin{align*}
\left\Vert f\right\Vert ^{2} & =\left|E\left(A,B\right)\right|c^{2}+\left|E\left(B,C\right)\right|a^{2}+\left|E\left(C,A\right)\right|b^{2}\\
 & \geq\left(\frac{k_{0}ab}{n}-\mu_{0}\sqrt{ab}\right)c^{2}+\left(\frac{k_{0}bc}{n}-\mu_{0}\sqrt{bc}\right)a^{2}+\left(\frac{k_{0}ca}{n}-\mu_{0}\sqrt{ca}\right)b^{2}\\
 & =k_{0}abc-\mu_{0}\left[\sqrt{ab}c^{2}+\sqrt{bc}a^{2}+\sqrt{ac}b^{2}\right]\geq k_{0}abc-\frac{\mu_{0}n^{3}}{9},
\end{align*}
and we are left with the task of bounding $\left\Vert \partial f\right\Vert ^{2}$.
Let us begin with
\begin{equation}
\begin{aligned}\sum_{\alpha\in A}\left(\partial f\right)^{2}\left(\alpha\right) & =\sum_{\alpha\in A}\Big(c\sum_{\beta\in B}\delta_{\alpha\beta}-b\sum_{\gamma\in C}\delta_{\alpha\gamma}\Big)^{2}\\
 & =c^{2}\sum_{{\alpha\in A\atop \beta,\beta'\in B}}\delta_{\alpha\beta}\delta_{\alpha\beta'}-2bc\sum_{\alpha,\beta,\gamma}\delta_{\alpha\beta}\delta_{\alpha\gamma}+b^{2}\sum_{\alpha,\gamma,\gamma'}\delta_{\alpha\gamma}\delta_{\alpha\gamma'}\\
 & =c^{2}\left|P\left(B,A,B\right)\right|-2bc\left|P\left(B,A,C\right)\right|+b^{2}\left|P\left(C,A,C\right)\right|,
\end{aligned}
\label{eq:trick}
\end{equation}
where $P\left(S,T,R\right)$ is the sets of paths of length two going
from $S$ to $R$ through $T$. By \cite[Lem.\ 1.3]{Parzanchevski2013}
with $\ell=2$ and $j=0$ (and also $k_{-1}=n$ and $\varepsilon_{-1}=0$),
we have
\begin{align*}
\left|\left|P\left(B,A,B\right)\right|-\left(\frac{k_{0}}{n}\right)^{2}b^{2}a\right| & \leq2k_{0}\mu_{0}b\\
\left|\left|P\left(B,A,C\right)\right|-\left(\frac{k_{0}}{n}\right)^{2}bac\right| & \leq2k_{0}\mu_{0}\sqrt{bc}\\
\left|\left|P\left(C,A,C\right)\right|-\left(\frac{k_{0}}{n}\right)^{2}c^{2}a\right| & \leq2k_{0}\mu_{0}c.
\end{align*}
Therefore,
\[
\sum_{\alpha\in A}\left(\partial f\right)^{2}\left(\alpha\right)\leq2k_{0}\mu_{0}\left[c^{2}b+2\left(bc\right)^{\frac{3}{2}}+b^{2}c\right]=2k_{0}\mu_{0}bc\left(\sqrt{b}+\sqrt{c}\right)^{2},
\]
and repeating this for $\sum_{\beta\in B}$ and $\sum_{\gamma\in C}$
gives
\[
\left\Vert \partial f\right\Vert ^{2}\leq2k_{0}\mu_{0}\left[bc\left(\sqrt{b}+\sqrt{c}\right)^{2}+ac\left(\sqrt{a}+\sqrt{c}\right)^{2}+ab\left(\sqrt{a}+\sqrt{b}\right)^{2}\right]\leq k_{0}\mu_{0}n^{3}.
\]
Combining everything together,
\begin{multline*}
\frac{\left|T\left(A,B,C\right)\right|n^{2}}{abc}\geq\frac{\lambda_{1}}{abc}\left(\left\Vert f\right\Vert ^{2}-\left\Vert f_{B}\right\Vert ^{2}\right)\geq\frac{\lambda_{1}}{abc}\left(\left\Vert f\right\Vert ^{2}\left(1-\frac{\mu_{0}}{k_{0}}\right)-\mu_{0}n^{3}\right)\\
\geq\frac{\lambda_{1}}{abc}\left(\left(k_{0}abc-\frac{\mu_{0}n^{3}}{9}\right)\left(1-\frac{\mu_{0}}{k_{0}}\right)-\mu_{0}n^{3}\right)\geq\lambda_{1}\left(k_{0}-\mu_{0}\left(1+\frac{10n^{3}}{9abc}\right)\right).
\end{multline*}

\end{proof}
This can be generalized to higher dimension. For a partition $A_{0},\ldots,A_{d}$
of the vertices of a $d$-complex, let $F\left(A_{0},\ldots,A_{d}\right)$
be the set of $d$-cells with one vertex in each $A_{i}$. We then
have:
\begin{thm}
\label{thm:cheeger-gen-dim}Let $X$ be a $d$-complex on $n$ vertices,
with $\Spec\Delta_{i}^{+}\big|_{Z_{i}}\subseteq\left[k_{i}-\mu_{i},k_{i}+\mu_{i}\right]$
for $0\leq i\leq d-2$ and $\Spec\Delta_{d-1}^{+}\big|_{Z_{d-1}}\subseteq\left[\lambda_{d-1},\infty\right)$.
Then for any partition $V=\coprod_{i=0}^{d}A_{i}$
\[
\frac{\left|F\left(A_{0},\ldots,A_{d}\right)\right|n^{d}}{\left|A_{0}\right|\cdot\ldots\cdot\left|A_{d}\right|}\geq k_{0}\cdot\ldots\cdot k_{d-2}\cdot\lambda_{d-1}\left(1-\frac{\mu_{d-2}}{k_{d-2}}-C_{d}\left(\frac{\mu_{0}}{k_{0}}+\ldots+\frac{\mu_{d-2}}{k_{d-2}}\right)\frac{n^{d+1}}{\prod_{i=0}^{d}\left|A_{i}\right|}\right),
\]
where $C_{d}$ depends only on $d$.
\end{thm}
Thus, if we define
\[
h_{\vartheta}\left(X\right)=\min\left\{ \frac{\left|F\left(A_{0},\ldots,A_{d}\right)\right|n^{d}}{\left|A_{0}\right|\cdot\ldots\cdot\left|A_{d}\right|}\,\middle|\,{V=\coprod_{i=0}^{d}A_{i}\atop \left|A_{i}\right|\geq\vartheta n}\right\}
\]
(where $0\leq\vartheta\leq\frac{1}{d+1}$), then
\[
h_{\vartheta}\left(X\right)\geq k_{0}\cdot\ldots\cdot k_{d-2}\cdot\lambda_{d-1}\left(1-\frac{\mu_{d-2}}{k_{d-2}}-\frac{C_{d}}{\vartheta^{d+1}}\left(\frac{\mu_{0}}{k_{0}}+\ldots+\frac{\mu_{d-2}}{k_{d-2}}\right)\right).
\]

For a complex with a complete skeleton, this reduces again to \cite[Thm.\ 1.2]{parzanchevski2012isoperimetric},
as $k_{i}=n$ and $\mu_{i}=0$ for $0\leq i\leq d-2$.
\begin{proof}
For $f\in\Omega^{d-1}\left(X\right)$ defined by
\[
f\left(\left[\sigma_{0}\:\sigma_{1}\:\ldots\:\sigma_{{d-1}}\right]\right)=\begin{cases}
\sgn\pi\left|A_{\pi\left({d}\right)}\right| & \exists\pi\in\mathrm{Sym}_{\left\{ 0\ldots{d}\right\} }\:\mathrm{with}\:\sigma_{i}\in A_{\pi\left(i\right)}\:\mathrm{for}\:0\leq i\leq{d-1}\\
0 & \mathrm{else,\: i.e.\:}\exists k,i\neq j\:\mathrm{with}\:\sigma_{i},\sigma_{j}\in A_{k},
\end{cases}
\]
it is shown in \cite[§4.1]{parzanchevski2012isoperimetric} that $\left\Vert \delta f\right\Vert ^{2}=\left|F\left(A_{0},\ldots,A_{{d}}\right)\right|n^{2}$
(this part does not use the complete skeleton assumption.) For $f_{B}=\mathbb{P}_{B^{d-1}}f$,
this gives again $\left|F\left(A_{0},\ldots,A_{{d}}\right)\right|n^{2}\geq\lambda_{d-1}\left(\left\Vert f\right\Vert ^{2}-\left\Vert f_{B}\right\Vert ^{2}\right)$.
Denoting $\mathcal{K}=k_{0}\cdot\ldots\cdot k_{d-2}$ and $\mathcal{E}=\frac{\mu_{0}}{k_{0}}+\ldots+\frac{\mu_{d-2}}{k_{d-2}}$,
we have by \cite[Thm.\ 1.1]{Parzanchevski2013}
\begin{align*}
\left\Vert f\right\Vert ^{2} & =\sum_{i=0}^{d}\left|F\left(A_{0},\ldots,\widehat{A_{i}},\ldots,A_{d}\right)\right|\left|A_{i}\right|^{2}\geq\sum_{i=0}^{d}\left[\frac{\mathcal{K}}{n^{d-1}}\smash{\prod_{j\neq i}}\left|A_{j}\right|-c_{d-1}\mathcal{K}\mathcal{E}\max_{j\neq i}\left|A_{j}\right|\right]\left|A_{i}\right|^{2}\\
 & \geq\frac{\mathcal{K}}{n^{d-2}}\prod_{i=0}^{d}\left|A_{i}\right|-\left(d+1\right)c_{d-1}\mathcal{K}\mathcal{E}n^{3}\geq\mathcal{K}\left(n^{2-d}\prod_{i=0}^{d}\left|A_{i}\right|-\left(d+1\right)c_{d-1}\mathcal{E}n^{3}\right).
\end{align*}
Turning to $f_{B}$, we have $\left\Vert f_{B}\right\Vert ^{2}\leq\frac{\mu_{d-2}}{k_{d-2}}\left\Vert f\right\Vert ^{2}+\frac{1}{k_{d-2}}\left\Vert \partial f\right\Vert ^{2}$
in a similar way to the triangle case, and we note that $\partial f$
is supported on $\left(d-2\right)$-cells with vertices in distinct
blocks of the partition $\left\{ A_{i}\right\} $. For a sequence
of sets $B_{0},\ldots,B_{\ell}$, denote by $F^{j}\left(B_{0},\ldots,B_{\ell}\right)$
the set of \emph{$j$-galleries} in $B_{0},\ldots,B_{\ell}$, namely,
sequences of $j$-cells $\tau_{i}\in F\left(B_{i},\ldots,B_{i+j}\right)$
such that $\tau_{i}$ and $\tau_{i+1}$ intersect in a $\left(j-1\right)$-cell.
Arguing similarly to \prettyref{eq:trick},
\begin{equation}
\begin{aligned}\left\Vert \partial f\right\Vert ^{2} & =\sum_{0\leq i<j\leq d\vphantom{\left(\widehat{A_{j}}\right)}}\:\sum_{\tau\in F\left(A_{0},\ldots,\widehat{A_{i}},\ldots,\widehat{A_{j}},\ldots,A_{d}\right)}\left(\partial f\right)\left(\tau\right)^{2}\\
 & =\sum_{0\leq i<j\leq d}\left|A_{j}\right|^{2}F^{d-1}\left(A_{i},A_{0},\ldots,\widehat{A_{i}},\ldots,\widehat{A_{j}},\ldots,A_{d},A_{i}\right)\\
 & \phantom{=\sum_{0\leq i<j\leq d}}-2\left|A_{i}\right|\left|A_{j}\right|F^{d-1}\left(A_{i},A_{0},\ldots,\widehat{A_{i}},\ldots,\widehat{A_{j}},\ldots,A_{d},A_{j}\right)\\
 & \phantom{=\sum_{0\leq i<j\leq d}}+\left|A_{i}\right|^{2}F^{d-1}\left(A_{j},A_{0},\ldots,\widehat{A_{i}},\ldots,\widehat{A_{j}},\ldots,A_{d},A_{j}\right).
\end{aligned}
\label{eq:gallery-counting}
\end{equation}
Proposition 3.1 in \cite{Parzanchevski2013} estimates of the number
of $j$-galleries in $B_{0},\ldots,B_{\ell}$ when each $j+1$ tuple
$B_{i},B_{i+1},\ldots,B_{i+j+1}$ consists of disjoint sets, giving
\[
\left|F^{d-1}\left(A_{i},A_{0},\ldots\widehat{A_{i}}\ldots\widehat{A_{j}}\ldots,A_{d},A_{i}\right)-\frac{\mathcal{K}k_{d-2}}{n^{d}}\left|A_{i}\right|\prod_{k\neq j}\left|A_{k}\right|\right|\leq c_{d-2,d}\mathcal{K}\mathcal{E}k_{d-2}\max_{k\neq j}\left|A_{k}\right|,
\]
and similarly for the other terms in \prettyref{eq:gallery-counting}.
Again, the main terms cancel out and the error terms result in $\left\Vert \partial f\right\Vert ^{2}\leq4\binom{d+1}{2}c_{d-2,d}\mathcal{K}\mathcal{E}k_{d-2}n^{3}$.
In total,
\begin{multline*}
\frac{\left|F\left(A_{0},\ldots,A_{d}\right)\right|n^{d}}{\left|A_{0}\right|\ldots\left|A_{d}\right|}\geq\frac{\lambda_{d-1}n^{d-2}}{\left|A_{0}\right|\ldots\left|A_{d}\right|}\left(\left(1-\frac{\mu_{d-2}}{k_{d-2}}\right)\left\Vert f\right\Vert ^{2}-\frac{1}{k_{d-2}}\left\Vert \partial f\right\Vert ^{2}\right)\\
\geq\mathcal{K}\lambda_{d-1}\left(\left(1-\frac{\mu_{d-2}}{k_{d-2}}\right)\left(1-\left(d+1\right)c_{d-1}\mathcal{E}\frac{n^{d+1}}{\prod\left|A_{i}\right|}\right)-4\binom{d+1}{2}c_{d-2,d}\mathcal{E}\frac{n^{d+1}}{\prod\left|A_{i}\right|}\right)\\
\geq\mathcal{K}\lambda_{d-1}\left(1-\frac{\mu_{d-2}}{k_{d-2}}-\left(\left(d+1\right)c_{d-1}+4\binom{d+1}{2}c_{d-2,d}\right)\frac{\mathcal{E}n^{d+1}}{\prod_{i=0}^{d}\left|A_{i}\right|}\right)
\end{multline*}
and the theorem follows.
\end{proof}

\section{\label{sec:pseudo-randomness}Pseudo-randomness}

In this section we use not only the lower bound on the edge spectrum,
but the fact that it is concentrated in two narrow stripes, to show
pseudo-random behavior for $2$-galleries. We first state a general
theorem, and then specialize to Ramanujan complexes.
\begin{thm}
\label{thm:mixing-general}Let $X$ be a regular tripartite triangle
complex with vertex and edge degrees $k_{0}$ and $k_{1}$, respectively,
such that
\begin{align*}
\Spec\Delta_{0}^{+}\big|_{Z_{0}} & \subseteq\left[k_{0}-\mu_{0},k_{0}+\mu_{0}\right]\cup\left\{ \tfrac{3k_{0}}{2}\right\} \qquad\mbox{and}\\
\Spec\Delta_{1}^{+}\big|_{Z_{1}} & \subseteq\left[k_{1}-\mu_{1},k_{1}+\mu_{1}\right]\cup\left[2k_{1}-\mu_{1},2k_{1}+\mu_{1}\right]\cup\left\{ 3k_{1}\right\} .
\end{align*}
If $A,B,C,D$ are disjoint sets of vertices of sizes $a,b,c,d$, respectively,
and each of $A\cup D$, $B$ and $C$ is contained in a different
block of the three-partition of $X$ (see Figure \ref{fig:2-gallery}),
then
\begin{equation}
\begin{aligned}\left|\left|F^{2}\left(A,B,C,D\right)\right|-\frac{27k_{0}k_{1}^{2}abcd}{2n^{3}}\right|\leq\frac{6\mu_{0}k_{1}^{2}\sqrt{abcd}}{k_{0}n}\left(\frac{3k_{0}\left(\sqrt{ab}+\sqrt{cd}\right)}{2n}+\mu_{0}\right)\\
+\left[\frac{2k_{1}^{2}\mu_{0}}{k_{0}}+\left(k_{1}+\mu_{1}\right)\mu_{1}\right]\sqrt[4]{abcd}\,\sqrt{\left(\tfrac{3k_{0}\sqrt{ab}}{2n}+\mu_{0}\right)\left(\tfrac{3k_{0}\sqrt{cd}}{2n}+\mu_{0}\right)}.
\end{aligned}
\label{eq:mixing}
\end{equation}

\end{thm}
The main term agrees with the pseudo-random expectation: given vertices
$\alpha,\beta,\gamma,\delta$ in $A,B,C,D$ respectively, the probability
that $\beta\gamma$ is an edge in $X$ is $\frac{3k_{0}}{2n}$, since
there are $\frac{k_{0}}{2}$ edges connecting $\beta$ with a vertex
from the block containing $C$, and $\frac{n}{3}$ such vertices.
Given that $\beta\gamma$ is in $X$, there are $k_{1}$ triangle
containing it and all have their third vertex in the block containing
$A\cup D$. The probability that $\alpha$ and $\delta$ are two of
these is $\frac{k_{1}\left(k_{1}-1\right)}{\nicefrac{n}{3}\left(\nicefrac{n}{3}-1\right)}$,
so that
\[
\mathbb{E}\left(\left|F^{2}\left(A,B,C,D\right)\right|\right)=\frac{3k_{0}}{2n}\cdot\frac{k_{1}\left(k_{1}-1\right)}{\nicefrac{n}{3}\left(\nicefrac{n}{3}-1\right)}\cdot abcd\approx\frac{27k_{0}k_{1}^{2}abcd}{2n^{3}}
\]

If $a,b,c,d\leq\vartheta n$ (where $\vartheta\leq\frac{1}{3}$) then
the l.h.s.\ in \prettyref{eq:mixing} is bounded by
\begin{multline*}
\left[\left(\frac{\left(6\vartheta+2\right)\mu_{{0}}}{k_{{0}}}+18\vartheta^{2}+3\vartheta\right)k_{{1}}^{2}\mu_{{0}}+\mu_{{1}}\left(\tfrac{3}{2}\vartheta k_{{0}}+\mu_{{0}}\right)\left(k_{{1}}+\mu_{{1}}\right)\right]\vartheta n\\
\leq\vartheta\left(9\vartheta+\frac{4\mu_{{0}}}{k_{{0}}}\right)\left(k_{{1}}\mu_{{0}}+k_{0}\mu_{{1}}\right)k_{1}n,
\end{multline*}
where in the second line we assume also $\mu_{1}\leq k_{1}$.

Ramanujan triangle complexes have $k_{0}=2\left(q^{2}+q+1\right)$,
$k_{1}=q+1$, $\mu_{0}=6q$ and $\mu_{1}=2\sqrt{q}$, and \prettyref{thm:Pseudo-random}
is obtained by applying this to \prettyref{eq:mixing} and using the
trivial bounds $q\geq2$ and $\vartheta\leq\frac{1}{3}$.

\smallskip{}

During the proof of the theorem we shall need a mixing lemma for colored
graphs, which we state now in broader generality: say that a graph
is $c$-colored $k$-regular if its vertices can be partitioned into
disjoint sets $V_{0},\ldots,V_{c-1}$ so that every $v\in V_{i}$
has no neighbors in $V_{i}$, and precisely $\frac{k}{c-1}$ neighbors
in each other $V_{j}$. The functions $f_{j}\left(V_{\ell}\right)\equiv\exp\left(\frac{2\pi ij\ell}{c}\right)/\sqrt{n}$
are orthonormal eigenfunctions of $\Delta_{0}^{+}$ with corresponding
eigenvalues
\begin{equation}
\lambda_{j}=\begin{cases}
0 & j=0\\
\left(\frac{c}{c-1}\right)k & 0<j<c,
\end{cases}\label{eq:colored-spectrum}
\end{equation}
and we call $\left\{ \lambda_{0},\ldots\lambda_{c-1}\right\} $ the
colored spectrum.
\begin{lem}
\label{lem:color-mixing}If the non-colored spectrum a $c$-colored
$k$-regular graph is contained in $\left[k-\mu,k+\mu\right]$, and
$A\subseteq V_{i}$, $B\subseteq V_{j}$ for $i\neq j$, then
\begin{equation}
\left|\left|E\left(A,B\right)\right|-\frac{ck\left|A\right|\left|B\right|}{\left(c-1\right)n}\right|\leq\mu\sqrt{\left|A\right|\left|B\right|}.\label{eq:colored-mixing}
\end{equation}
\end{lem}
\begin{proof}
This is a straightforward generalization of the standard lemma. Assuming
that $A\subseteq V_{0}$ and $B\subseteq V_{1}$, and denoting by
$\mathbb{P}_{W}$ the orthogonal projection on $W=\left\langle f_{0},\ldots,f_{c-1}\right\rangle ^{\bot}$,
we have
\begin{align*}
\left|E\left(A,B\right)\right| & =\left\langle \left(kI-\Delta_{0}^{+}\right)\one_{A},\one_{B}\right\rangle \\
 & =\sum_{j=0}^{c-1}\left(k-\lambda_{j}\right)\left\langle \one_{A},f_{j}\right\rangle \left\langle \one_{B},f_{j}\right\rangle +\left\langle \left(kI-\Delta_{0}^{+}\right)\mathbb{P}_{W}\one_{A},\one_{B}\right\rangle \\
 & =\frac{\left|A\right|\left|B\right|}{n}\sum_{j=0}^{c-1}\left(k-\lambda_{j}\right)\exp\left(\frac{2\pi ij}{c}\right)+\left\langle \left(kI-\Delta_{0}^{+}\right)\mathbb{P}_{W}\one_{A},\one_{B}\right\rangle ,
\end{align*}
and \prettyref{eq:colored-mixing} follows by \prettyref{eq:colored-spectrum}
and $\left\Vert \left(kI-\Delta_{0}^{+}\right)\big|_{W}\right\Vert \leq\mu$.
\end{proof}

\begin{proof}[Proof of \prettyref{thm:mixing-general}]
Denote by $U^{+}$ the space spanned by the $\Delta_{1}^{+}$-eigenforms
with eigenvalue in $\left[k_{1}-\mu_{1},k_{1}+\mu_{1}\right]\cup\left[2k_{1}-\mu_{1},2k_{1}+\mu_{1}\right]$,
and by $\eta$ the normalized disorientation
\[
\eta\left(e\right)=\sqrt{\frac{2}{nk_{0}}}\left(-1\right)^{\col e},
\]
so that $\Omega^{1}\left(X\right)=B^{1}\oplus U^{+}\oplus\left\langle \eta\right\rangle $.
If $p\left(x\right)=\left(x-k_{1}\right)\left(x-2k_{1}\right)$, then
$p\left(\Delta_{1}^{+}\right)$ acts on $B^{1}\oplus\left\langle \eta\right\rangle $
as the scalar $2k_{1}^{2}$, and
\[
\left\Vert p\left(\Delta_{1}^{+}\right)\big|_{U^{+}}\right\Vert \leq\max\left\{ \left|p\left(\lambda\right)\right|\,\middle|\,\lambda\in\begin{smallmatrix}\left[k_{1}-\mu_{1},k_{1}+\mu_{1}\right]\,\cup\,\\
\left[2k_{1}-\mu_{1},2k_{1}+\mu_{1}\right]
\end{smallmatrix}\right\} =\left(k_{1}+\mu_{1}\right)\mu_{1}.
\]
Let us express $p\left(\Delta_{1}^{+}\right)$ in combinatorial terms:
we say that two directed edges $e,e'$ are neighbors if they have
a common origin or a common terminus, and their union (as an unoriented
cell) is in $X^{2}$. We denote this by $e\sim e'$, and by $\mathcal{A}$
the corresponding adjacency operator on $\Omega^{1}\left(X\right)$,
namely $\left(\mathcal{A}f\right)\left(e\right)=\sum_{e'\sim e}f\left(e'\right)$.
The upper Laplacian satisfies
\[
\Delta_{1}^{+}=k_{1}\cdot I-\mathcal{A},
\]
(see \cite{parzanchevski2012isoperimetric,Parzanchevski2013}), and
it follows that
\[
p\left(\Delta_{1}^{+}\right)=\mathcal{A}^{2}+k_{1}\mathcal{A}.
\]
Define $\one_{AB}\in\Omega^{1}\left(X\right)$ by
\[
\one_{AB}\left(vw\right)=\begin{cases}
1 & v\in A,w\in B\\
-1 & v\in B,w\in A\\
0 & otherwise
\end{cases}
\]
and similarly $\one_{CD}$. We claim that
\begin{equation}
\left|F^{2}\left(A,B,C,D\right)\right|=\left\langle p\left(\Delta_{1}^{+}\right)\one_{AB},\one_{CD}\right\rangle .\label{eq:two-galleries}
\end{equation}
Indeed, edges in $E\left(A,B\right)$ have no neighbors in $E\left(C,D\right)$,
so that $\left\langle \mathcal{A}\one_{AB},\one_{CD}\right\rangle =0$,
and $\mathcal{A}$ is self-adjoint (since $\Delta_{1}^{+}$ is), giving
\begin{align}
\left\langle p\left(\Delta_{1}^{+}\right)\one_{AB},\one_{CD}\right\rangle  & =\left\langle \mathcal{A}^{2}\one_{AB},\one_{CD}\right\rangle =\left\langle \mathcal{A}\one_{AB},\mathcal{A}\one_{CD}\right\rangle \nonumber \\
 & =\sum_{{e,e',e''\in X^{1}\atop e'\sim e\sim e''}}\one_{AB}\left(e'\right)\one_{CD}\left(e''\right).\label{eq:sum_1AB_1CD}
\end{align}
The nonzero terms in this sum come from edges $e$ which have neighbors
$e'\in E\left(A,B\right)$ and $e''\in E\left(C,D\right)$, so they
must be in one of $E\left(A,C\right)$, $E\left(A,D\right)$, $E\left(B,C\right)$
or $E\left(C,D\right)$. In fact, only $E\left(B,C\right)$ is possible:
$E\left(A,D\right)$ is empty as $A$ and $D$ are of the same color;
if $e\in E\left(A,C\right)$ then $e\sim e''$ implies that $e\cup e''$
is a triangle in $T\left(A,C,D\right)$ which is again impossible,
and similarly for $E\left(B,D\right)$ using $e'$. Thus, $\left(e'\cup e,e\cup e''\right)$
is a $2$-gallery in $F^{2}\left(A,B,C,D\right)$, and its contribution
to \prettyref{eq:sum_1AB_1CD} is always one: if $e$ goes from $B$
to $C$, it shares its origin with $e'$ and its terminus with $e''$,
so that $\one_{AB}\left(e'\right)=\one_{CD}\left(e''\right)=1$, and
if $e$ goes from $C$ to $B$ then $\one_{AB}\left(e'\right)=\one_{CD}\left(e''\right)=-1$.
On the other hand, for every gallery $\left(t,t'\right)\in F^{2}\left(A,B,C,D\right)$,
the edges $e'=t\backslash C$, $e=t\cap t'$, $e''=t\backslash B$
form such a triplet, and we obtain \prettyref{eq:two-galleries}.

On the spectral side we have
\[
\left\langle p\left(\Delta_{1}^{+}\right)\one_{AB},\one_{CD}\right\rangle =2k_{1}^{2}\left\langle \mathbb{P}_{B^{1}\oplus\left\langle \eta\right\rangle }\one_{AB},\one_{CD}\right\rangle +\left\langle p\left(\Delta_{+}^{1}\right)\mathbb{P}_{U^{+}}\one_{AB},\one_{CD}\right\rangle ,
\]
and the last term is bounded by
\begin{equation}
\left\Vert p\left(\Delta_{+}^{1}\right)\big|_{U^{+}}\right\Vert \left\Vert \one_{AB}\right\Vert \left\Vert \one_{CD}\right\Vert \leq\left(k_{1}+\mu_{1}\right)\mu_{1}\sqrt{E_{AB}E_{CD}},\label{eq:mu1-error}
\end{equation}
where $E_{AB}=\left|E\left(A,B\right)\right|$ and and $E_{CD}=\left|E\left(C,D\right)\right|$.
As $\eta$ has constant sign on $V_{0}\rightarrow V_{1}\rightarrow V_{2}\rightarrow V_{0}$,
\begin{equation}
2k_{1}^{2}\left\langle \mathbb{P}_{\left\langle \eta\right\rangle }\one_{AB},\one_{CD}\right\rangle =2k_{1}^{2}\left\langle \one_{AB},\eta\right\rangle \left\langle \eta,\one_{CD}\right\rangle =\frac{4k_{1}^{2}E_{AB}E_{CD}}{k_{0}n},\label{eq:disor-contrib}
\end{equation}
and we are left to analyze $\mathbb{P}_{B^{1}}\one_{AB}$. As in the
non-tripartite case, one has $\Spec\Delta_{-}^{1}\big|_{B^{1}}=\Spec\Delta_{+}^{0}\big|_{B_{0}}$,
but now the latter comprises not only eigenvalues in $\left[k_{0}-\mu_{0},k_{0}+\mu_{0}\right]$,
but also $\frac{3k_{0}}{2}$ (twice, see \prettyref{thm:ram-triangle-spec}).
If $\omega=\exp\left(\tfrac{2\pi i}{3}\right)$ and
\[
\xi\left(vw\right)=\left\{ \begin{aligned}\sqrt{\nicefrac{2}{k_{0}n}} &  & v\in V_{0},w\in V_{1} &  &  &  &  & -\sqrt{\nicefrac{2}{k_{0}n}} &  & w\in V_{0},v\in V_{1}\\
\omega\sqrt{\nicefrac{2}{k_{0}n}} &  & v\in V_{1},w\in V_{2} &  &  &  &  & -\omega\sqrt{\nicefrac{2}{k_{0}n}} &  & w\in V_{1},v\in V_{2}\\
\overline{\omega}\sqrt{\nicefrac{2}{k_{0}n}} &  & v\in V_{2},w\in V_{0} &  &  &  &  & -\overline{\omega}\sqrt{\nicefrac{2}{k_{0}n}} &  & w\in V_{2},v\in V_{0},
\end{aligned}
\right.
\]
then $\left\{ \xi,\overline{\xi}\right\} $ is an orthonormal basis
for the $\frac{3k_{0}}{2}$-eigenspace of $\Delta_{1}^{-}$. Denote
by $U^{-}$ the space spanned by the $\Delta_{1}^{-}$-eigenforms
with eigenvalue in $\left[k_{0}-\mu_{0},k_{0}+\mu_{0}\right]$. By
the action of each summand in
\[
\mathfrak{D}'\overset{{\scriptscriptstyle def}}{=}k_{0}\mathbb{P}_{B^{1}}+\frac{k_{0}}{2}\mathbb{P}_{\left\langle \xi,\overline{\xi}\right\rangle }-\Delta_{1}^{-}
\]
on each of the terms in the orthogonal decomposition $\Omega^{1}\left(X\right)=Z_{1}\oplus U^{-}\oplus\left\langle \xi,\overline{\xi}\right\rangle $
we see that $\left\Vert \mathfrak{D}'\right\Vert \leq\mu_{0}$. Due
to the fact that $\partial_{1}\one_{AB}$ and $\partial_{1}\one_{CD}$
are supported on different vertices, $\left\langle \Delta_{1}^{-}\one_{AB},\one_{CD}\right\rangle $
vanishes, and together with
\[
\left\langle \mathbb{P}_{\left\langle \xi,\overline{\xi}\right\rangle }\one_{AB},\one_{CD}\right\rangle =2\Re\left(\left\langle \one_{AB},\xi\right\rangle \left\langle \xi,\one_{CD}\right\rangle \right)=-\frac{2E_{AB}E_{CD}}{k_{0}n}
\]
(and $\mathbb{P}_{B^{1}}=\frac{1}{k_{0}}\Delta_{1}^{-}-\frac{1}{2}\mathbb{P}_{\left\langle \xi,\overline{\xi}\right\rangle }+\frac{1}{k_{0}}\mathfrak{D}'$)
this gives
\[
\left|2k_{1}^{2}\left\langle \mathbb{P}_{B^{1}}\one_{AB},\one_{CD}\right\rangle -\frac{2k_{1}^{2}E_{AB}E_{CD}}{k_{0}n}\right|\leq\frac{2k_{1}^{2}}{k_{0}}\left\langle \mathfrak{D}'\one_{AB},\one_{CD}\right\rangle \leq\frac{2\mu_{0}k_{1}^{2}\sqrt{E_{AB}E_{CD}}}{k_{0}}.
\]
Combining this with \prettyref{eq:mu1-error} and \prettyref{eq:disor-contrib}
we conclude that
\[
\left|\left|F^{2}\left(A,B,C,D\right)\right|-\frac{6k_{1}^{2}E_{AB}E_{CD}}{k_{0}n}\right|\leq\left(\frac{2\mu_{0}k_{1}^{2}}{k_{0}}+\left(k_{1}+\mu_{1}\right)\mu_{1}\right)\sqrt{E_{AB}E_{CD}}.
\]
This estimates $\left|F^{2}\left(A,B,C,D\right)\right|$ in terms
of $E_{AB}$ and $E_{CD}$. To have an estimate in terms of $a,b,c$
and $d$ we use \prettyref{lem:color-mixing}, which gives $\left|E_{AB}-\frac{3k_{0}ab}{2n}\right|\leq\mu_{0}\sqrt{ab}$
and similarly for $E_{CD}$, and the theorem follows.
\end{proof}
As an application of our ``$2$-gallery mixing lemma'' we give a
lower bound for the weak chromatic number of non-tripartite Ramanujan
complexes. This is the minimal number of colors needed to color $X^{0}$
so that every triangle has vertices of at least two different colors.
For example, the weak chromatic number of a tripartite triangle complex
is two, as one can merge two of the blocks in a three-partition into
a single color. If $X=\Gamma\backslash\mathcal{B}_{3}$ is non-tripartite,
recall that $\widehat{\Gamma}$ (see \prettyref{eq:Gamma-hat}) is
a normal subgroup of $\Gamma$ of index three, and $\widehat{X}=\widehat{\Gamma}\backslash\mathcal{B}_{3}$
is a tripartite three-cover of $X$. Let $V_{0}\amalg V_{1}\amalg V_{2}$
be the partition of of $\widehat{X}^{0}$ according to $\col=\ord_{\pi}\det$.
If the chromatic number of $X$ is $\chi$, we can find a set $N\subseteq X^{0}$
of size $\frac{n}{\chi}$ with $T\left(N,N,N\right)=\varnothing$
(any monochromatic set would satisfy this). Let $N_{0},N_{1},N_{2}$
be the preimages of $N$ in $V_{0},V_{1}$ and $V_{2}$, respectively,
and take $A=N_{0}$, $B=N_{1}$, $C=N_{2}$ and $D=V_{0}\backslash N_{0}$.
\prettyref{thm:mixing-general} then estimates $\left|F^{2}\left(A,B,C,D\right)\right|$
with main term
\[
\frac{27k_{0}k_{1}^{2}abcd}{2\left(3n\right)^{3}}\geq\frac{q^{4}abcd}{n^{3}}=\frac{q^{4}n\left(\chi-1\right)}{\chi^{4}}\geq\frac{q^{4}n}{2\chi^{3}}
\]
and an error term bounded by
\[
\left(\frac{84q^{3.5}}{\chi^{1.5}}+\frac{119q^{2.5}}{\sqrt{\chi}}+139q^{1.5}\right)n.
\]
However, $F^{2}\left(A,B,C,D\right)$ must be empty, since any triangle
in $T\left(A,B,C\right)$ would cover one in $T\left(N,N,N\right)$,
hence the error term is at least $\frac{q^{4}n}{2\chi^{3}}$. This
implies $\chi\geq\frac{\sqrt[3]{q}}{30}$: otherwise, $\chi\leq\frac{\sqrt[3]{q}}{30}$
leads to
\[
\frac{\sqrt{q}}{2\chi^{1.5}}\leq\left(84+\frac{119\chi}{q}+\frac{139\chi^{1.5}}{q^{2}}\right)\leq87
\]
which implies again $\chi\geq\frac{\sqrt[3]{q}}{30}$. This bound
is better than $\frac{\sqrt[6]{q}}{2}$ which is obtained in \cite{evra2014mixing}
using use only the action of Hecke operators on vertices. Note however
that the results in \cite{evra2014mixing} do not assume the complex
to be Ramanujan, and apply to higher dimensions as well.

Other applications of a triangle mixing lemma can be adjusted, in
a similar manner, to use our gallery mixing lemma. This includes,
for example, Gromov's overlap property, along the lines of \cite{FGL+11,Parzanchevski2013},
and the crossing numbers of complexes, as discussed in \cite[§8.1]{Gundert2013}.
Nevertheless, the question of triangle pseudorandomness remains interesting,
and should give better results if it does hold. The fact that most
of the spectrum is concentrated in the strip $\mathcal{I}$ (see \prettyref{thm:ram-triangle-spec})
gives hope that this can be done by analyzing the combinatorics of
eigenforms which occur in the principal series (type \prettyref{enu:prin-unit}
in \prettyref{sec:Unitary-Iwahori-spherical-repres}), and showing
that their contribution is negligible.

\bibliographystyle{amsalpha}
\bibliography{mybib}

\bigskip{}

\parbox[t]{0.55\columnwidth}{
\noun{Einstein Institute of Mathematics}

\noun{The Hebrew University of Jerusalem }

\noun{Jerusalem, 91904, Israel}

E-mail: \texttt{kost.golubev@mail.huji.ac.il}
}
\parbox[t]{0.44\columnwidth}{
\noun{School of Mathematics}

\noun{Institute for Advanced Study}

\noun{Princeton, NJ 08540, USA}

E-mail: \texttt{parzan@ias.edu}
}
\end{document}