text:
\begin{document}

\title[Infinite Fibering]{Hyperbolic Groups Which Fiber in Infinitely Many Ways}

\author[T. Mecham, A. Mukherjee]{TaraLee Mecham, Antara Mukherjee}
\address{Dept.\ of Mathematics\\
        University of Oklahoma\\
        Norman, OK 73019\\
        USA}
\email{tmecham@math.ou.edu, amukherjee@ou.edu}
\date{}

\begin{abstract}
We construct examples of free-by-cyclic hyperbolic groups which
fiber in infinitely many ways over $\Z$.  The construction involves adding a
specialized square 2-cell to a non-positively curved, squared
2-complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are hyperbolic, free-by-cyclic and can be mapped onto $\Z$ in infinitely many ways.
\end{abstract}

\maketitle

\section{Introduction}
Topologists have been interested in fiberings of 3-manifolds since
the early 1960's. A 3-manifold is said to fiber over $S^1$ if it
is the mapping torus of a surface via a homeomorphism. A key to
relating this topological fibering to the notion of algebraic
fibering was expressed by Stallings in 1962 \cite{St}. He
proved:
   \emph{For an irreducible compact 3-manifold, $M^3$, if
   $\pi_1(M^3)$ has a finitely generated, normal subgroup G (which is not $\Z_2$), whose quotient group is
   $\Z$, then $M^3$ fibers over $S^1$. The fiber here is a 2-manifold $T$ embedded in $M^3$, such that $\pi_1(T)= G$. }
    The fundamental groups of these manifolds are a special class of HNN-extensions that extend the fundamental group of the surface by $\Z$.

     In 1969, Tollefson showed that there were 3-manifolds that fibered over $S^1$ in
infinitely many ways, \cite{To}. From the group theory
perspective, this means that the fundamental groups of  these
3-manifolds can be expressed in infinitely many ways as
surface-by-cyclic groups with the genus of the surface going to
infinity. In 1998 Thurston \cite{Wi}
 proved that \emph{every atoroidal three-manifold that
fibers over the circle has a hyperbolic structure}.  Using branched covers of the 3-torus, Hilden, Lozano
and Montesinos-Amilibia \cite{HiLo}
     constructed the first explicit examples of hyperbolic manifolds which have infinitely many fibrations over
$S^1$, in 2006.
 The main theorem of this paper produces
analogous results in the context of hyperbolic, free-by-cyclic
groups.

\begin{theorem}
\label{main} There exist hyperbolic, free-by-cyclic groups which
have  infinitely many epimorphisms onto $\Z$. Furthermore, each of
these groups is isomorphic to $F_n \rtimes \Z$ for infinitely many
$n$.
 \end{theorem}
 The groups we produce will be the first concrete examples of hyperbolic, free-by-cyclic groups which fiber in infinitely many ways over $\Z$.  We prove Theorem \ref{main} by constructing groups
that are hyperbolic, free-by-cyclic and map onto $\Z^2$. These
groups are the fundamental groups of specially constructed
non-positively curved, squared 2-complexes.

 The paper is organized as follows:
 Section 2 gives some
background about non-positively curved squared 2-complexes,
$\delta$-hyperbolic spaces, and Morse Theory  necessary for the
construction of the groups. Section 3 gives a description of the
building blocks for the construction in Section 4. We
give the actual construction of groups with reducible monodromy automorphism,  proving Theorem \ref{main},  in Section 4.
Section 5 provides an alternative construction to produce an
example of a group which may have an irreducible automorphism.
This construction does not use the building blocks from Section 3.  We conclude the paper with
Section 6, where we present some questions that are yet to be
answered.

This paper would not have been possible without the support and guidance of our advisor, Dr. Noel Brady.

\section{Background and Definitions}

Theorem \ref{main} requires that we produce free-by-cyclic, hyperbolic groups, so we will recall some facts about
$\delta$-hyperbolic spaces and combinatorial Morse Theory in this section.
 Since the  groups we construct are the fundamental groups of
non-positively curved, squared 2-complexes, we first recall some
facts about non-positively curved spaces.  Then we quote Gromov's
Flat Plane Theorem in \cite{Gr} which helps us show that
the groups are indeed hyperbolic.
 Finally, we will present
some facts about Morse Theory and conclude the section with
Proposition \ref{FbyC} which we will use
to both prove that our groups are free-by-cyclic, and compute the free rank of the groups.

We start with some definitions associated with non-positively curved spaces.
These definitions can be found in Bridson and
Haefliger's  book \cite{BrHa}.

  \begin{definition}[Piecewise  2-complex]
  \label{PE2cmplx}
    A \emph{piecewise Euclidean 2-complex} $K$ is
    a 2-complex obtained from a collection of convex cells in the Euclidean plane
    by identifying their edges via isometries.
\end{definition}

  \begin{definition}[Link of a vertex]
\label{link}
 The link of a vertex $v$ in a convex 2-cell $c$
(denoted by $Lk(v,c)$) of a 2-complex, is defined as the
collection of unit tangent vectors to the space at $v$ which point
to $c$. This is a spherical cell of dimension $(n-1)$.
  \end{definition}

Suppose there are two convex 2-cells $c, c'$ with vertices $v, v'$
respectively which are identified along the faces via an isometry
$f$ which identifies $v$ with $v'$.  The derivative of $f$ gives a
spherical isometry which identifies a face of $Lk(v,c)$ and a face
of $Lk(v',c')$. Thus, the the link of a vertex in a 2-complex is
naturally described as a piecewise spherical complex.

  \begin{definition}[Link Condition]
  \label{Linkcon}
    The link of a vertex satisfies the
     \emph{large link condition} if it has no circuits of length $< 2\pi $.  This definition is the same as saying that  the link
     satisfies the no empty triangles condition or equivalently,  the link is a flag
     complex.   See \cite{BrHa} for more details.

  \end{definition}

  \begin{definition}[Non-positively curved 2-complex]
  \label{NPC2cmplx}
     A piecewise Euclidean 2-complex $K$ is said to be \emph{non-positively curved} if $Lk(v,K)$ is large (i.e., no circuits of length less than $2\pi$) for all $v \in K^{(0)}$.
      In this context a link is a simplicial complex, which implies it has no edge loops and no bigons.
  \end{definition}

  The following is a theorem stated by Gromov  \cite{Gr}, which was later proved by Bridson
\cite{Br}.  We will use this theorem to show that the universal
covers of the 2-complexes we construct are hyperbolic, which will
imply that the corresponding fundamental groups are hyperbolic. This
result is analogous to  Eberlein's \cite{Eb} earlier work, where
he showed that:  \emph{A compact non-positively curved Riemannian
manifold either has a word-hyperbolic fundamental group or its
universal cover contains an isometrically embedded copy of
the Euclidean plane}.

  \begin{theorem}[The Flat Plane Theorem \cite{BrHa}]
  \label{Hyp}
     A proper cocompact CAT(0) space is \emph{hyperbolic} if and only if it does not contain a subspace
     isometric to $\mathbb E^2$.
  \end{theorem}

 We will also need to
ensure that our groups are free-by-cyclic.  We will use
combinatorial Morse Theory to establish this fact.  Following are
the basic definitions necessary for this method of proof.  These
definitions, and proofs for the theorems, can be found in \cite{BaBr}.

  \begin{definition}[Affine cell-complex]
  \label{Affine}
    A finite-dimensional cell-complex $X$ is said to be an \emph{affine cell-complex} if it has the following
    structure.  An integer $ m > dim(X)$ is given, and for each cell $e$ of $X$ we are given a convex
    polyhedral cell $C_e \subset$ $\mathbb R^m$ and a characteristic function $\chi_e:C_e\rightarrow e$
    such that the restriction of $\chi_e$ to any face of $C_e$ is a characteristic function of another cell,
    possibly pre-composed by a partial affine homeomorphism of $\mathbb R^m$.
  \end{definition}

  In our construction, our highest dimensional cell will be two-dimensional.  In fact, the complexes will be  made of Euclidean squares identified
  along the edges, so they will be affine.

     \begin{definition}[Morse Function]
  \label{Morse}
    A map $f:X \rightarrow \mathbb R$ defined on an affine cell-complex $X$ is a \emph{Morse
function}
    if:

        \begin{itemize}
             \item
                 for every cell $e$ of $X$, $f_{\chi_e} :C_e \rightarrow \mathbb R$ extends to an
affine map $
                 \mathbb R^m \rightarrow \mathbb R$ and $f_{\chi_e}$ is constant only when $dim(e)
= 0$, and

            \item
                the image of the $0$-skeleton is discrete in $\mathbb R$.

        \end{itemize}
  \end{definition}

  One can think of a Morse function as a height function where each cell of the complex has  a unique maximum and a
  unique minimum.  Since we want to show that our groups are free-by-cyclic, we will be interested in maps
  from our complexes to the circle.  The following definition relates these maps to Morse functions.

   \begin{definition}[Circle-valued Morse function]
   \label{CircleMorse}
      A \emph{circle-valued Morse function} on an affine cell complex $X$ is a cellular map $f:X
\rightarrow
      S^1$, with the property that $f$ lifts to a Morse function between universal covers.  When the context is clear, we will simply refer to these as Morse functions.

   \end{definition}

  The links of the vertices of a 2-complex are graphs.
  It turns out that in the presence of a Morse function,  certain subgraphs of the link give us important information about the structure of the
   fundamental group of the 2-complex.  The following definition and proposition describe this relationship.

  \begin{definition}[Ascending and descending links]
  \label{AscDesLinks}
     Suppose $X$ is an affine cell complex and $f:X \rightarrow S^1$ is a circle-valued Morse
function.
     Choose an orientation of $S^1$,  which lifts to one of $\mathbb R$, and lift $f$ to a map of
universal
     covers $\widetilde{f}:\widetilde{X}\rightarrow \mathbb R$.  Let $v \in X^{(0)} $, and note
that the link of
      $v $ in $X$ is naturally isomorphic to the link of any lift $\widetilde{v}$ of $v$ in
$\widetilde{X}$.  We
      say that a cell $\widetilde{e}\subset \widetilde{X}$ contributes to the \emph{ascending}
(respectively
      \emph{descending}) link of $\widetilde{v}$ if $ \widetilde{v} \in \widetilde{e}$ and only if
$\widetilde{f}|
      _{\widetilde{e}}$ achieves its minimum (respectively maximum) value at $\widetilde{v}$.  The
      ascending (respectively descending) link of $v$ is then defined to be the subset of
$Lk(v,X)$
      naturally identified with the ascending (respectively descending) link of $\widetilde{v}$.

  \end{definition}

The following proposition is an important result of Morse Theory and
helps us check that the fundamental groups of the 2-complexes we
construct are indeed free-by-cyclic and gives us a formula for determining the free rank of the groups.
  \begin{proposition}[Free-by-Cyclic \cite{BaBr}]
  \label{FbyC}
     If $f:X \rightarrow S^1$ is a circle-valued Morse function on the 2-complex X all of whose
ascending
     and descending links are trees, then X is aspherical, and $\pi_1(X)$ is free-by-cyclic.  In
fact, $\pi_1
     (X) = F_k \rtimes \Z$ where $k = 1 - \chi (f^{-1}(p))$ for $p\in S^1$.
 \end{proposition}

 \section{Building Blocks}
\label{blocks}

The groups that we construct
 in Section 4 are one-relator quotients of free
products of some building block groups.  In this section we describe
these building block groups and show that they are the fundamental
groups of non-positively curved 2-complexes.  We also show that these
groups are hyperbolic and free-by-cyclic.

\begin{remark}  We will be looking at 2-complexes which have a
unique vertex for the rest of the paper.
\end{remark}

We start this section by defining a Labeled Oriented Graph (LOG).
The group presentation corresponding to a particular LOG is used as the building block group for the example
constructed in section 4.  The following definition is taken from
\cite{BaBr}.

  \begin{definition}[Labeled Oriented Graph]
  \label{LOG}
    A \emph{labeled oriented graph} or LOG, consists of a finite, directed graph with distinct
labels on all
    the vertices and oriented edge labels taken from the set of vertex labels.  An LOG defines a
finite
    group presentation in the following manner.  The set of generators is in 1-1 correspondence
with the
    set of vertex labels and the set of relators is in 1-1 correspondence with the set of edges,
so that an
    edge labeled \emph{a} oriented from vertex \emph{u} to vertex \emph{v} corresponds to a
conjugation
    relation $ava^{-1}=u$.  In the case that the graph is a tree,  we call it a \emph{labeled
oriented tree} or
    LOT, and call the corresponding finite presentation an LOT presentation.
  \end{definition}

 Howie \cite{Ho} was one of the first people to define and study the group
presentation of LOTs.

\begin{figure}[htbp] \begin{center} \input{lotandlink.eps_t} \caption{The LOT presentation of $L_{(a_i)}$  and the link of $v$ in $X_{L_{(a_i)}}$.  The circle in the center of the link represents the points $4^+$ and $4^-$.  The dashed lines are edges connecting to $4^-$.  There is no edge connecting $4^+$ to $4^-$.    The descending link is shown in bold.} \label{lotandlink} \end{center} \end{figure}

  \begin{remark}
    The main component of our construction is an LOT group with five generators
$a_0,...,a_4$
    denoted by $L_{(a_i)}$.   Figure \ref{lotandlink} shows the graph of the LOT.
The corresponding presentation 2-complex  $X_{L_{(a_i)}}$, consists of  a bouquet of five
circles and four square 2-cells. The group presentation of this complex is given by,
  $$
    L_{(a_i)} = \langle a_0,a_1,a_2,a_3,a_4 \hspace{0.2cm}|\hspace{0.2cm} a_4 = a_0^{a_1},\hspace
    {0.2cm} a_4 = a_1 ^{a_2},\hspace{0.2cm} a_4 = a_2 ^{a_3},\hspace{0.2cm}a_3 = a_4^{a_0}\rangle
  $$
where $a_i^{a_j}= a_ja_ia_j^{-1}$.

The link of the vertex of the corresponding presentation 2-complex is also shown in Figure \ref{lotandlink}.  $X_{L_{(a_i)}}$ is a piecewise Euclidean 2-complex and it is easy to
see that the link is large and hence, the universal cover
$\widetilde{X}_{L_{(a_i)}}$ is CAT(0). In order to show that
$L_{(a_i)}$ is hyperbolic we will need to use the Flat Plane
Theorem.   The following definition will be useful in its
application.

\end{remark}

\begin{definition}[Poison Corners]
\label{poison} A \emph{poison corner} is a corner of a 2-cell for
which the corresponding edge in the link is not a part of any
length-four circuit.
\end{definition}

\begin{remark}
This definition is helpful to us because the 2-cells of the
presentation 2-complex $X_{L_{(a_i)}}$ are Euclidean square 2-cells.
This means that all edges in the link are of length $\pi/2$. Hence,
any square of our complex that contains a poison corner cannot be
used to develop a flat plane in the universal cover.
\end{remark}

  \begin{theorem}
  \label{XLAI}
    The universal cover of  $X_{L_{(a_i)}}$ has no isometrically embedded flat planes.
  \end{theorem}

\begin{proof}

    The link has no bigons
or empty
   triangles, so the link is large.  We now need only ensure that no planes develop in the universal cover.
    There are four edges of the link that act as bridges from the inner
suspension to
   the outer circuit of the link.  They are:
  $$
       \hspace{0.5cm}a_1^{+} a_0^{+},\hspace{0.5cm}a_2^{+} a_1^{-},\hspace{0.5cm} a_3^{+} a_2^{-},
       \hspace{0.5cm}a_0^{-} a_3^{-}
  $$
 If we remove these four edges, the link reduces to two separate components.  The outer component is the length-four circuit ${a_0}^-
{a_1}^+{a_2}^+{a_3}
   ^+{a_0}^-$.  The combinatorial distance between any two vertices of this component is at least
1.  On
   the other hand, the inner component is the suspension on the two vertices ${a_4}^+$ and
${a_4}^- $
   that connecting them to ${a_0}^+,{a_1}^-,{a_2}^-,$ and ${a_3}^-$.
   The distance between any pair of these four vertices in the inner component is at least 2.  Therefore, any
circuit
   containing one of these bridges has to be of length at least 5.  Hence, the bridges represent poison corners.

   Any isometrically embedded flat plane is tiled by a collection of 2-cells of the presentation
2-complex
   $X_{L_{(a_i)}}$. However, every 2-cell of this complex contains a poison corner.  This means that there are no
isometrically
   embedded flat planes in the universal cover of the complex.
   \end{proof}

  \begin{corollary}
  \label{LAIisHYP}
     $L_{(a_i)}$ is a hyperbolic group.
   \end{corollary}
\begin{proof}
 The universal cover of the 2-complex $X_{L_{(a_i)}}$ is CAT(0) since the link is large and Theorem
\ref{XLAI} further
   implies that the universal cover $\widetilde{X}_{L_{(a_i)}}$ has no isometrically embedded flat
planes, so using the Flat Plane Theorem, we know that $\widetilde{X}_{L_{(a_i)}}$ is $\delta$-hyperbolic and the corresponding group $L_{(a_i)}$ is hyperbolic.
  \end{proof}

Now we will show that $L_{(a_i)}$ is also free-by-cyclic. The
proof of the following proposition is along the same lines as the first part of the proof of Theorem 3 in \cite{BaBr}.

\begin{proposition}
$L_{(a_i)}$ is free-by-cyclic and is isomorphic to $F_4\rtimes \Z$.
\end{proposition}
\begin{proof}
  The presentation 2-complex $X_{L_{(a_i)}}$ of
the group $L_{(a_i)}$, admits a circle-valued Morse function which
is defined by mapping the vertex to the base vertex of the circle
and each directed edge is mapped once around the target circle. The
same map extends linearly over the 2-cells of
$X_{L_{(a_i)}}$.
 The descending link of this Morse function is shown in bold on the link in Figure
 \ref{lotandlink}.
Since both the ascending and descending links are trees, Proposition
\ref{FbyC} implies that the LOT group is free-by-cyclic. The
pre-image of the vertex in the target circle under the Morse
function is a bouquet of four circles, so
the group is isomorphic to  $F_4\rtimes \Z$.
\end{proof}

Although $L_{(a_i)}$ is hyperbolic and free-by-cyclic, it only
fibers in one way over $\Z$ because the relations of the group are
all conjugation relations.  This means that the image of all the
generators will be the same under any map to $\Z$. Therefore there
is only one nontrivial map to $\Z$, up to post composition with a
monomorphism of $\Z$.

\begin{remark}
\label{otherLOT}

It is easy to see that for any LOT group represented by a tree
with $k+1$ vertices, a single valence $k$ vertex, and with edge
and vertex labels permuted as in the $L_{(a_i)}$ case, the
previous theorems hold. In fact, the link will consist of an
outer $k$-gon with $k$ bridges to $k$ inner vertices and the
suspension of these inner vertices on the vertices $a_k^+$ and
$a_k^-$.  This means that the group will be hyperbolic and
isomorphic to $F_k\rtimes \Z$. For arbitrary $k \geq 4$, we will
denote the group and the presentation 2-complex  associated to it
by $L_{(k, a_i)}$, and $X_{L_{(k, a_i)}}$ respectively.

\end{remark}

\section{The Construction of Reducible Examples}
\label{const}

Now we will use the LOT groups from the previous section to prove
Theorem \ref{main}.  We will begin by focusing on the topological
aspects of our construction and show that the groups produced
are free-by-cyclic and map onto $\Z^2$. This will be the content
of Theorem \ref{maps} below.
 Next we will address the geometry of our particular examples by
showing that they are indeed hyperbolic, thus providing the proof
of Theorem \ref{main}.  We will then give a method for computing
the free rank of the kernel of a map from our groups onto $\Z$ in
Lemma \ref{growth}. We conclude this section by writing down
an explicit (reducible) automorphism for the group produced in the
proof of Theorem \ref{main}.
 .

  \begin{remark}
  \label{Notation}
Since all of the spaces that we deal with have a single vertex, we shall denote the fundamental
group
     element corresponding to any 1-cell say $x$, of a 2-complex, by
$[x]$.
If $f$ is the map from a presentation 2-complex, $K$, to a circle
and $f_*$ the map between the corresponding fundamental groups,
i.e, $f_*:\pi _1(K)\rightarrow \Z$, then using the notation given above,
we will define $\alpha = f_*([a])$ and $\beta = f_*([b])$.  We will use this notation throughout the rest of the paper.

  \end{remark}

   \begin{theorem}
   \label{maps}
     Given 2-complexes $K_j$ for $j= 1,2$ satisfying:

        \label[(i)   $|K_j^{(0)}|= 1$,

        \label[(ii)  there exist Morse functions $ f_j: K_j \rightarrow S^1$ such that the
ascending and
        descending links are trees,

        \label[(iii)  there are 1-cells $a_0,a_1 \in K_1$ and $b_0, b_1 \in K_2$ such that
$|\alpha_0 |+
        |\beta_1| = |\alpha_1| + |\beta_0|$,

     then the 2-complex, $K$, obtained from ${K_1 \sqcup K_2}/(K_1^{(0)} \sim K_2^{(0)})$ by
attaching a
     square 2-cell according to the relation $r=a_0^{\pm 1}b_1^{\pm 1}a_1^{\mp1}b_0^{\mp1}$, satisfies:

        \label[(i)   $\pi _1(K)$ is free-by-cyclic and,

        \label[(ii)   $\pi _1(K)$ maps onto $\Z^2$.

   \end{theorem}

\begin{proof}
   \label[(i)
      We start by labeling the unique vertex of $K_j$ by $v_j$ for $j = 1, 2$ and consider the
2-complex  $K_1
      \vee K_2 = {K_1 \sqcup K_2}/v_1 \sim v_2$, calling the common vertex $v$.  The fundamental
group
      of  this complex is the free product of the fundamental groups of $K_1$ and $K_2$.
     By the functorality of the wedge product, $f_1$ and $f_2$ induce the following map:
         $$ f_1\vee f_2 : K_1 \vee K_2 \rightarrow S^1\vee S^1$$
     Using $f_1\vee f_2$, we define a circle-valued Morse function, $pr\circ (f_1\vee f_2)$, on the 2-complex $K_1\vee K_2$ so that $pr\circ (f_1\vee f_2)\circ incl. = f_j$ for $j=1,2$, where $pr$ is the projection map of $S^1\vee S^1$ onto $S^1$. Following is a commutative diagram showing the relationship between these maps.
           \begin{displaymath}
             \xymatrix{K_j \ar[d]_{f_j}\ar@{^{(}->}[r]^{incl.}  &  K_1\vee K_2 \ar[d]^{f_1\vee
f_2} \\ S^1        &       S^1
             \vee S^1\ar[l]^{pr}}
        \end{displaymath}

  The ascending and descending links of $v$ are each 2-component
forests
  (the union of the ascending and descending links of $v_1\in K_1$ and $v_2 \in K_2$).  To
complete the
  construction, we attach a single 2-cell which represents a relation $ r $ of the form $a_0^{\pm 1}b_1^{\pm 1}=
b_0^{\pm 1}a_1^{\pm 1}$ where these are the 1-cells described in
hypothesis  \label[(iii).  The powers in the relation are positive
or negative depending upon the sign of the associated $\alpha _ i$,
$\beta_i$.   Since $\alpha_0 + \beta_1 =  \alpha_1 + \beta_0$, we
can use Lemma 1.5 of Scott and Wall's paper \cite{ScTe} to extend
the Morse function of $K_1\vee K_2$
  over $K$.  We will call this new function $g$.  The extra 2-cell contributes an edge each to the ascending and descending
  links connecting up the two components of the forest in each case.  Thus the ascending and
  descending links are now trees and $\pi_1(K) $ is free-by-cyclic by Proposition \ref{FbyC}.  The
  following commutative diagram shows the Morse function on the complex $K$.

     \begin{displaymath}
          \xymatrix{K_1 \vee  K_2 \ar[d]_{f_1\vee f_2} \ar@{^{(}->}[r]  & K \ar@{.>}[d]^{g}\\
     S^1\vee S^1 \ar[r] _{pr}                             & S^1 }
    \end{displaymath}

\label[(ii)There is also a map from $K_1\vee K_2$ to $S^1\times S^1$ which is shown in the next diagram.  SInce $K_1\vee K_2$ is a
  sub-complex of $K$ we can use the same result as above in \cite{ScTe} and extend this map to a map $g'$ over $K$.

       \begin{displaymath}
        \xymatrix{K_1 \vee  K_2 \ar[d]_{f_1\vee f_2} \ar@{^{(}->}[r]  & K \ar@{.>}[d]^{g'}\\
         S^1\vee S^1  \ar@{^{(}->}[r]   & S^1\times S^1 }
     \end{displaymath}

  This gives us a map  from $\pi_1(K)= \pi_1(K_1) \ast \pi_1(K_2)/<<r>>$ to $\Z^2$.  The map to $\Z^2$ is
  onto since it is composed of $f_{1_*}: \pi_1(K_1) \rightarrow \Z$ and $f_{2_*}: \pi_1(K_2) \rightarrow \Z
  $, where the maps  $f_{j_*}$  are defined by sending generators of $\pi_1(K_j)$ to generators of each
  factor of $\Z$ .

  \end{proof}

  Now all the pieces are in place for us to provide the proof of the main theorem.

   \begin{proof}[Proof of Theorem \ref{main}]
  \label{LOT-LOT}

    We begin with 2 copies of our LOT group namely, $L_{(a_i)}$, $L_{(b_i)}$ and their associated
    presentation 2-complexes, $X_{L_{(a_i)}}$, $X_{L_{(b_i)}}$.  Using the above construction, we
obtain a new complex $X_1$ such that its fundamental group is:
       $$G_1 = L_{(a_i)}\ast L_{(b_i)}/\langle \langle a_0b_2a_1^{-1}b_0^{-1}\rangle \rangle  =
\langle L_
       {(a_i)}, L_{(b_i)}\hspace{0.2cm} | \hspace{0.5cm}a_0b_2a_1^{-1}b_0^{-1} \rangle $$
  Any choice of relation $a_ib_j = b_ka_l$ would fit the conditions of Theorem \ref{maps}, but not
all of
  these would ensure that our resultant group is hyperbolic, so we have carefully chosen
indices for the generators in the relation so that the link of the
new complex $X_1$ is large and the corresponding universal cover
  contains no flat planes.  Figure \ref{lotlotlink} represents the link $X_1$.  As we can see, there are no new circuits
of length less than five created in the link when the edges
corresponding to the extra square 2-cell  are added.  Thus, $X_1$
is non-positively curved and the poison corners from each original
complex are still present in $X_1$ prohibiting the development of
a flat plane in the universal cover, $\widetilde{X_1}$. Therefore
$\widetilde{X_1}$ is $\delta$-hyperbolic, and $G_1$ is a
hyperbolic group.  Theorem \ref{maps} implies our group is
free-by-cyclic and  there is a map from $G_1$ onto $\Z^2$, thus
completing the proof of Theorem \ref{main}.

   \end{proof}

\begin{figure}[htbp] \begin{center} \input{lotlotlink.eps_t} \caption{$Link(v,X_1)$.  The descending link is shown in bold.} \label{lotlotlink} \end{center} \end{figure}

   We leave it to the reader to check that $\langle L_{(k_1, a_i)}, L_{(k_2, b_i)}\hspace{0.2cm} | \hspace{0.4cm}a_0b_2a_1^{-1}b_0^{-1} \rangle$ is also free-by-cyclic and hyperbolic for all $k_1, k_2  \geq 4$.
    The following lemma gives us a formula for calculating the free rank of the kernel when there is an epimorphism from these groups onto $\Z$.

 \begin{figure}[htbp] \begin{center} \input{growthcell.eps_t} \caption{Example 2-cells of $K$ for the case $m=2$ and $n=3$, with the edges sub-divided.  The dashed lines represent the  pre-image of $p$ in these 2-cells.} \label{growthcell} \end{center} \end{figure}

   \begin{lemma}
   \label{growth}

Let  $K$ be a 2-complex constructed as in Theorem \ref{maps} such that
$K_j = X_{L_{(a_i,k_j)}}$ for $k_1, k_2 \geq 4$ and the added 2-cell reads $ a_0^{\pm1}b_2^{\pm1}a_1^{\mp1}b_0^{\mp1}$ around the boundary.
  Let $f_j:K_j \rightarrow S^1$, $j=1,2$, be circled-valued Morse functions such that  $m =|{f_1}_*(a_i)|$ and $n=|{f_2}_*(b_i)|$, and let $g:K\rightarrow S^1$ be the circle valued Morse function obtained from the construction in Theorem \ref{maps}.  Then, if $m, n \neq 0$ and $gcd(m, n) = 1$,  $\chi(g^{-1}(p)) =-(k_1m+k_2n)$.

     \end{lemma}

  \begin{proof}  First we will examine the point pre-images of the map $f_j:K_j
\rightarrow S^1$. Since we are dealing with affine cell-complexes
and affine maps, these pre-images are graphs.
 Since $K_j = X_{L_{(a_i, k)}}$,  $K_j$ is made up of $k$ square 2-cells with $(k+1)$ different edge
labels.
  For $|f_{j_*}(a_i)| = m$, the pre-image of the basepoint $p$  of $S^1$ is the union of its pre-images in the
2-cells.  Each
  edge will have to be subdivided which will add $(m-1)$ new vertices for each edge label (i.e. $(k+1)(m-1)
+1$
  total vertices).  The number of edges in the pre-image in a single 2-cell is equal to the number
of pairs of
  opposing vertices in the 2-cell, which is $2(m-1) + 1$.
  The left-hand side of Figure \ref{growthcell} shows a sample cell for the case $m=2$. The resulting graph has Euler characteristic:
$$(k+1)(m-1) +1 -
  k(2(m-1) +1)= -(k-1)m.$$

 Now we will look at the pre-image in the added 2-cell.  (The case where $m=2$ and $n=3$ is shown in Figure \ref{growthcell}.)  We will not count vertices since they
are counted in the building block complexes.  The number of edges
is $(m+n-1)$ which is again the number of opposing pairs of vertices in the cell.

Since the 2-complex $K$ is composed of $X_{L_{(k_1, a_i)}}$,
$X_{L_{(k_2, b_i)}}$ and the added 2-cell, after making the correction for the over count
of the original vertex by subtracting 1, we get:  $$\chi(g^{-1}(p)) = -(k_1-1)m
+ -(k_2 -1)n  - (m+n-1) - 1 = -(k_1m + k_2n).$$
\end{proof}

\begin{remark}
For $gcd(m,n)= d >1$ let $m'=m/d, n'=n/d$; then
$gcd(m',n')=1$.  Using lemma \ref{growth} we have
$\chi(h^{-1}(p)) = 1 -d(k_1m'+k_2n')-d$ where $h$ is the
composition of the circle-valued Morse function $g$ in the lemma
above with a  map which is multiplication by $d$ at the fundamental
group level i.e,
$h_* = (\times d)\circ g_*$.
\end{remark}

\begin{remark}
From the above result we see that the free rank of any group
constructed in this manner is $(k_1m + k_2n +1)$.  The minimal free rank
of $G_1$ above is therefore 9, and using these methods, we can
construct a group with any minimal free rank $k\geq 9$.
\end{remark}

Finally we claim that the examples we obtained in this section so
far have reducible automorphisms associated with them.
 Bestvina and Handel \cite{BeHa} gave the following definition of reducible and
 irreducible automorphisms.

\begin{definition}
     An automorphism of a group G is \emph{reducible} if it preserves the conjugacy class of a proper free
      factor of G.  If the automorphism is not reducible then it is \emph{irreducible}.

\end{definition}

The main example produced in this section is the group
$F_9\rtimes\Z$. This group was constructed as a one-relator
quotient of a free product of two free-by-cyclic groups (both of
free rank 4). We will show that there is a reducible automorphism
associated with this group.

 \begin{figure}[htbp] \begin{center} \input{reducible.eps_t} \caption{The generators of $F_9$.  The generators are labeled according to the edge labels of the corresponding 2-cell which appear twice in the cell.} \label{reducible} \end{center} \end{figure}

\begin{proposition}
\label{reducible} There exists a reducible automorphism associated
with the group $F_9\rtimes\Z$ constructed in the example above.
\end{proposition}

\begin{proof}

 Consider the pre-image of the
base-point of the target circle
 under the Morse function $g: X \rightarrow S^1$ (from Theorem \ref{maps}). This pre-image
 can be described as a union of the pre-images under $f_1:X_{L_{(a_i)}} \rightarrow S^1$ and
 $f_2:X_{L_{(b_i)}} \rightarrow S^1$ along with the pre-image in the extra 2-cell .
  The pre-images under $f_1$ and  $f_2$ are bouquets of 4 circles.
  Let $\alpha_0,.....,\alpha_3$ be the generators of the
 free group (denoted by $F_{\alpha_i}$) corresponding to the pre-image under
 $f_1$. Similarly let $\beta_0,....,\beta_3$ be the generators of the free group (denoted by
$F_{\beta_i}$) corresponding to the pre-image
 under $f_2$. Also, let the generator corresponding to the pre-image in the
 remaining 2-cell be $\gamma$.  Figure \ref{reducible} shows generic square
 2-cells of the 2-complex $X$ and the edge running through the middle of each square 2-cell representing the generators $\alpha_i, \beta_{i'}$ and $\gamma$.

\begin{figure}[htbp] \begin{center} \input{autos.eps_t} \caption{The effect of conjugating by $a_0$ on the generators $\alpha_1$ and $\beta_3$.  The automorphism is given by reading the labels on the bottom of the figures.} \label{autos} \end{center} \end{figure}

 If we conjugate by a generator from the
group $L_{(a_i)}$, say $a_0$, then $F_{\alpha_i}$ is invariant.
This can be shown by examining what effect this conjugation has on the generators.
$$ a_0 \alpha_0
{a_0}^{-1} = \alpha_3^{-1} \alpha_2^{-1} \alpha_1^{-1},  \hspace{0.3cm} a_0
\alpha_1 {a_0}^{-1} = \alpha_1 {\alpha_0},$$
$$a_0
\alpha_2 {a_0}^{-1} = \alpha_1 \alpha_2 {\alpha_0},
\hspace{0.3cm} a_0 \alpha_3 {a_0}^{-1} = \alpha_1 \alpha_2
\alpha_3 \alpha_0 $$
 Figure \ref{autos} illustrates how these relations are obtained.
 The automorphism
corresponding to the above conjugation actually keeps $F_{\alpha_i}$ invariant.  Thus this automorphism is reducible. We
can also see that $F_{\beta_i}$ is not invariant, but it is invariant up to conjugation.   Conjugating by $a_0$, gives us $\alpha \gamma^{-1} F_{\beta_i} (\alpha
\gamma^{-1})^{-1}$ as seen below.
$$a_0 \beta_0 a_0^{-1} = \alpha_1 \gamma^{-1} \beta_2^{-1} \beta_1^{-1}
\beta_3^{-1} \gamma \alpha_1^{-1}, \hspace{0.3cm} a_0 \beta_1 a_0^{-1}=
\alpha_1 \gamma^{-1} \beta_2^{-1} \beta_0 \beta_1 \beta_2 \gamma
\alpha_1^{-1}$$
$$a_0 \beta_2 a_0^{-1}= \alpha_1 \gamma^{-1} \beta_0 \beta_1
\beta_2 \gamma \alpha_1^{-1},\hspace{0.3cm} a_0 \beta_3 a_0^{-1} =
\alpha_1\gamma^{-1} \beta_3 \beta_0 \beta_1 \beta_2 \gamma \alpha_1^{-1}$$
In fact, conjugating by any $a_i$ generator will keep $F_{\alpha_i}$ invariant just as conjugating by any $b_i$ generator will keep $F_{\beta_i}$ invariant.  If we conjugate by other elements that map to $1\in \Z$, we will still preserve the conjugacy classes of these factors.  For example:
$$ (b_0 a_1^{-1}a_0) F_{\beta_i}( a_0^{-1} a_1 b_0^{-1}) \cong (b_0a_1^{-1})\alpha \gamma^{-1} F_{\beta_i}\gamma \alpha^{-1}(a_1b_0^{-1})$$
Since $b_0a_0^{-1}$ is in the kernel of $g_*$, we see that the conjugacy class of $F_{\beta_i}$ is still preserved under conjugation in $F_9 \rtimes \Z$.  Thus any automorphism will be reducible.

\end{proof}

This same technique can be used to give reducible automorphisms  for any group of the form, $\langle L_{(k_1, a_i)}, L_{(k_2,
b_i)}\hspace{0.2cm} | \hspace{0.4cm}a_0b_2a_1^{-1}b_0^{-1}
\rangle$, for $k_1, k_2 \geq 4$.

\section{A Labeled Oriented Forest Example }
By construction, the examples produced in the previous section are necessarily reducible.  In this section we will use a different method of construction to produce an example that does not immediately appear to be reducible.  This construction is similar to that presented in Section 4, but instead of taking a free product of two separate groups and adding a relation, we will simply add a relation to an already existing group.
The group we will begin with will be defined from a Labeled Oriented Forest (LOF).

\begin{figure}[htbp] \begin{center} \input{lof.eps_t} \caption{The LOF corresponding to the group $G_F$.} \label{lof} \end{center} \end{figure}

\label{LOF1}
\begin{example}
\label{LOF} (Labeled Oriented Forest):  Our LOF will consist of
the two trees shown in Figure \ref{lof}.  The first tree has
vertices labeled by $a_1, . . ., a_4$ and the second has vertices
labeled by $b_1, . . ., b_4$.  The edges of each tree are labeled by
vertices of the opposite tree.  The presentation 2-complex $X$ of
this LOF, is comprised of six square 2-cells and admits a
circle-valued Morse function $f: X\rightarrow S^1$.   The ascending
and descending links are 2-component forests.
 The corresponding group is: $$G_F = \langle a_i, b_i, i=1..4\hspace{0.3cm}| \hspace{0.3cm}
a_1^{b_2} = a_4,\hspace{0.1cm} a_2^{b_3} = a_4,
\hspace{0.1cm}a_4^{b_1} = a_3,\hspace{0.1cm} b_1^{a_2} =
b_4,\hspace{0.1cm} b_2^{a_3} =b_4,\hspace{0.1cm} b_4^{a_1} = b_3
\rangle . $$

The link of the complex (see Figure \ref{loflinks}) has no bigons or
empty triangles, therefore the link is large.  No flat planes
develop in the universal cover because every 2-cell of our complex
has not just one, but two poison corners.

We complete the
construction by adding an extra square 2-cell that represents the relation $a_4b_1 =
b_4a_1$.  The Morse function $f$ extends to a map $g_2$ over
the new complex which we
 will call $X_2$.  The corresponding group $G_2$ is denoted by:  $\langle G_F\hspace{0.3cm}|\hspace{0.3cm} a_4b_1a_1^{-1}b_4^{-1}\rangle$.

 The ascending and descending links corresponding to the new complex are now trees as the new
 relation contributes one edge each to both links joining up the 2-components of the forests.  Hence, the corresponding group is free-by-cyclic by Proposition \ref{FbyC}.
   The addition of this cell has added length 4 circuits to our link, so we must now check the new link to show
   that $G_2$ is hyperbolic.  The new link is shown on the right  in Figure
   \ref{loflinks}. The four edges contributed by the new
   2-cell are denoted here by dashed-dotted lines between the following vertices:
   ${b_4}^+{a_1}^-$, ${b_4}^-{a_4}^-$,  ${a_4}^+{b_1}^-$, and
   ${a_1}^+{b_1}^+$.
   We can see that where we previously had twelve poison corners, we now have only  four.
    Consequently,  our new 2-cell, and two of our original 2-cells no longer have poison corners.
    The following lemma will show that $G_2$ is indeed hyperbolic.

\begin{figure}[htbp] \begin{center} \input{loflinks.eps_t} \caption{$Link(X, v)$ and $Link(X_2, v)$.  The circles indicate suspension points as in Figure \ref{lotandlink}.  Poison Corners are indicated by bold edges.  Note that although the one dot-dash edges and the two dot-dash edges appear to meet at the suspension points, they do not.  A careful check shows that there are no circuits of length less than four. } \label{loflinks} \end{center} \end{figure}

\begin{lemma}
   $G_2$ is hyperbolic.
\end{lemma}

 \begin{proof}  In order to prove this lemma, we must check that the universal cover of $X_2$ is $\delta$-hyperbolic,
 i.e., we must check that no flat planes can develop in the universal cover $\tilde{X_2}$ .
 Any such plane would have to be made up of the new square 2-cell, and only those original 2-cells which now have no poison
 corners.  We will attempt to build a plane starting with the new cell and show
 that it is impossible to develop a flat plane, thus proving the lemma.
 The 2-cells available to us are shown at the top of Figure \ref{poison}.
 In the bottom of Figure \ref{poison}, we see that it does not take long to discover a vertex around which we cannot find a Euclidean neighborhood
 (indicated by the large arrow).

 \end{proof}

\begin{figure}[htbp] \begin{center} \input{poison.eps_t} \caption{The 2-cells of $X_2$ which have no poison corners fail to develop into a plane in the universal cover.} \label{poison}\end{center} \end{figure}

 $G_2$ can be mapped onto $\Z^2$ because the $a_i$ generators and $b_i$ generators can be sent independently to generators of $\Z^2$.  Hence, $G_2$ fibers in infinitely many ways over $\Z$.
 The following lemma investigates the ranks of the kernels of these maps.
  \begin {lemma}
   Let $G_2$ and $g_2$ be as in Example \ref{LOF}.  Suppose that $m = {g_2}_*(a_i)$ and $n={g_2}_\ast(b_i)$ for $i=1,. .,4$.  Then if $m, n \neq 0$ and $gcd(m,n)= 1$, the rank of the kernel of the map ${g_2}_\ast : G_2 \rightarrow Z$ is given by the following expression:  $1+(3|m|+3|n|)$.
\end{lemma}

 \begin{proof}  By construction,  $m>0$ implies $n>0$, so we will assume without loss of generality that $m,n > 0$. In order
to find the rank,
  we will examine the Euler characteristic of the graph which is the pre-image of the base-point
of
  $S^1$ under the map $g_2$.   Since the 2-complex contains seven 2-cells, we will
  look at what the effect of changing the map $g_{2_\ast} $ has on each 2-cell.  In the presentation 2-complex, all the 2-cells look the same.  For $m=1$ and $n=1$, the pre-image has one vertex and seven edges (one from each 2-cell), so
the
   rank of the kernel is 7.
   Otherwise we must subdivide the edges, adding new vertices.   The number of new vertices corresponding to
each
   $a_i$, $b_i$ edge is  $4(m-1),4(n-1)$ respectively.   The number of edges in each 2-cell is $(m+n-1)$  (see the right-hand side Figure \ref{growthcell}),  so in all
there are $7(m+n-1)$ edges.  Counting the original vertex, the
total number of vertices is $4m+4n-7$, so the  Euler characteristic
of the graph is $-(3m+3n)$.  Therefore the rank of the free kernel is
$1+3m+3n$.

   \end{proof}

   \end{example}

   \begin{remark}For the example constructed in this section, we can show that there is a
  topological representative of an outer automorphism which is irreducible.  We will show this using definitions and techniques from Bestvina-Handel's paper \cite{BeHa}, Section 1.

  Let $R_n$ be the bouquet on $n$-circles.  A marked graph $G$ is a
graph with a homotopy equivalence $\tau: R_n \rightarrow G$.  A
homotopy equivalence $f: G \rightarrow G $ determines an outer
automorphism $ \mathcal{O}$, of $F_n$. If $\textit{V}$ is the
vertex set of $G$ and $f|_{ (G\setminus \textit{V})}$ is locally
injective, then we say that $f: G \rightarrow G$ is a
\emph{topological representative} of $ \mathcal{O}$.
 The transition matrix $M$ for a topological representative $f$ has entries $a_{ij}$ defined as the
number of times that the $f$-image of the $j ^{th}$ edge crosses
the $i^{th}$ edge in either direction.  A nonnegative integral matrix $M$ is irreducible if for all $1
\leq i, j \leq dim(M)$, there exists $N(i,j)>0$ so that the
$ij^{th}$ entry of $M^{N(i,j)}$ is positive.  A topological representative, $f$, is irreducible if and
only if its transition matrix is irreducible.  An outer automorphism $\mathcal{O}$ is irreducible if and only if every topological representative of $\mathcal{O}$ is irreducible.

Consider  the map ${g_2}_*: G_2 \rightarrow \Z$ such that $1 =
{g_2}_*(a_i), 1 = {g_2}_*(b_i)$.  Using the same
naming convention as in Proposition \ref{reducible}, let the generators of the
Ker(${g_2}_*$) be $\alpha_1, \alpha_2, \alpha_3, \beta_1, \beta_2,
\beta_3, \gamma$.  If we conjugate the generators by $a_3$ say, then we have the following
expressions:
$$ a_3 \alpha_1{a_3}^{-1} = \alpha_3^{-1} \beta_2^{-1} \beta_3^{-1}
\alpha_2^{-1}\gamma \beta_2 \alpha_3,
 \hspace{0.4cm} a_3
\alpha_2 {a_3}^{-1} = \alpha_3^{-1} \beta_2^{-1} \gamma^{-1} \alpha_2
\alpha_1\beta_2\alpha_3,$$
$$ a_3 \alpha_3 {a_3}^{-1} = \alpha_1 \beta_2 \alpha_3,
 \hspace{0.35cm}a_3 \beta_1
{a_3}^{-1} = \alpha_3^{-1} \beta_2^{-1} \gamma^{-1},
\hspace{0.35cm} a_3
\beta_2 {a_3}^{-1} = \alpha_3^{-1} \beta_1 \gamma \beta_2
\alpha_3,$$
$$a_3\beta_3{a_3}^{-1} = \alpha_3^{-1} \beta_2^{-1} \gamma^{-1}
\alpha_2 \beta_3 \beta_1 \gamma \beta_2
\alpha_3,
\hspace{0.4cm} a_3 \gamma {a_3}^{-1} = \alpha_3^{-1}
\beta_2^{-1} \gamma^{-1} \beta_1^{-1} \alpha_1\beta_2 \alpha_3$$
This gives us a topological representative of an outer
automorphism $\mathcal{O}$ and the corresponding transition matrix
$M$ is a $7 \times 7$ nonnegative matrix.  Using applicable software, it is easy to calculate the powers of the is matrix and see that
all entries of $M^3$ are positive.  Therefore, we have an
irreducible topological representative of the outer automorphism; however, using this method it is not possible to check that every topological
representative is irreducible.
Gersten and Stallings \cite{GeSt} provide a method for showing that an outer automorphism is irreducible, using super irreducible characteristic polynomials of $\mathcal{O}_{ab}$.  This method also failed to give a positive result due to the abelianization process involved.

   \end{remark}

   \section{Questions}
Following are some open questions concerning hyperbolic, free-by-cyclic groups that fiber in infinitely many ways.

 \emph{Question 1.}  We know that the minimal rank of
free kernels for hyperbolic, free-by-cyclic groups is 3.  Using our methods we can generate groups that fiber in infinitely many ways with free kernel of minimal rank $7$ and
any minimal rank $k \geq 9$.  Can we
find lower rank examples that fiber in infinitely many ways?

The impetus for this research was work done in the 3-manifold setting.  The next question relates to results obtained in that setting.

\emph{Question 2.} Every hyperbolic 3-manifold that fibers in infinitely many ways has an irreducible outer automorphism.  Is the automorphism in the LOF example
 irreducible?  Are there any other examples with irreducible automorphisms?

All of our examples have maps onto $\Z^2$.  A group is said to be large if a finite index subgroup maps onto $F_2$.  Our next question deals with possible connections to large groups.

\emph{Question 3.}  Are there free-by-cyclic groups that fiber in only one way, such as $L_{(a_i)}$, which have finite index subgroups which fiber in infinitely
many ways?  Will these groups have virtual first Betti number $> 1$.   Would these groups be large?

\bibliography{hyperfiber17}
\bibliographystyle{siam}

\end{document}

config: arxiv


TITLE: Ratio of sum of squares of Normal Distributions
QUESTION [3 upvotes]: Find $$E[\frac{\sum_{i=1}^{51} X_i^2}{\sum_{i=51}^{101} X_i^2}], $$if $X_1,X_2,...,X_{51}$ are independent and are distributed as N(0,1). This question appeared in my final exam of probability and stochastic processes, that occurred today .I couldn't answer this in the exam and would love to find out more. I did simplify this a bit though. Here's what I did:
$Y = \sum_{i=1}^{50} X_i^2$, $Z = \sum_{i=52}^{101} X_i^2,X = X_{51}$. Then,
$X \sim \gamma(1/2,1/2), Y \sim \gamma(25,1/2), Z \sim\gamma(25,1/2). $ Then we wish to find $E[\frac{X+Y}{X+Z}]$, where X,Y,Z are independent random variables. I was stuck after this, as there was this integral of $$\frac{x^{-1/2}}{x+z}e^{-25x}, 0\leq x < \infty$$ Any help will be appreciated.

REPLY [2 votes]: You can write the ratio as $\frac{Y}{X+Z} + \frac{X}{X+Z}$. Then $\frac{Y}{X+Z}$ has an F-distribution with $d_1 = 50$ and $d_2 = 51$ degrees of freedom. And $\frac{X}{X+Z}$ has a Beta-distribution with parameters $\alpha = \frac{1}{2}, \beta = \frac{50}{2} = 25$ .


TITLE: $f$ be a strictly monotonic continuous function on $[a,b]$
QUESTION [2 upvotes]: Let $f$ be a strictly monotonic continuous function defined on $[a,b$ such that $f(a)<a$ and $f(b)>b$. Then there exists exactly one $c\in (a,b)$ duch that $f(c)=c.$

Define $g(x)=f(x)-x$. Clearly, $g$ is a continuous function. $g(a)=f(a)-a<0$ and $g(b)=f(b)-b>0$. So using Intermediate value theorem, we can say that there exists a $c\in (a,b)$ such that $g(c)=0,$ i.e. $f(c)=c$. But why only one $c$?

REPLY [5 votes]: It is not true.  Let $f(x)=x^3$ on $[-2,2]$  $f(x)=x$ is true for $-1,0,1$


TITLE: If $T(t)$ is a semigroup, why is $t\mapsto T(t)x$ Riemann integrable?
QUESTION [2 upvotes]: Let

$E$ be a $\mathbb R$-Banach space
$(T(t))_{t\ge0}$ be a semigroup on $E$
$x\in E$ with $$\left\|T(h)x-x\right\|_E\xrightarrow{h\to0+}0\tag1$$ (i.e. $(T(t))_{t\ge0}$ is strongly continuous at $x$)

In many books, we can find that $$\frac1h\int_t^{t+h}T(s)x\:{\rm d}s=T(t)x\tag2\;\;\;\text{for all }t\ge0,$$ but why does the Riemann integral on the left-hand side of $(2)$ exist at all?

By $(1)$, the only thing we shoulbe be able to conclude is that $$\operatorname{orb}x:[0,\infty)\to E\;,\;\;\;t\mapsto T(t)x$$ is right-continuous, but in order to show that $(2)$ holds, it seems like that we need that $\operatorname{orb}x$ is continuous. As far as I'm not mistaken, this cannot generally be shown.

What am I missing?

EDIT: By $(1)$, we easily see that $\operatorname{orb}x$ is right-continuous, since if $t\ge0$, then $$\|(\operatorname{orb}x)(t+h)-(\operatorname{orb}x)(t)\|_E\le\|T(t)\|_{\mathfrak L(E)}\|T(h)x-x\|_E\tag4$$ for all $h\ge0$.
But I don't think we can show left-continuity (without a local boundedness assumption). If $t\ge0$, then $$T(t)=T(t-h)T(h)\tag6$$ and hence $$\|(\operatorname{orb}x)(t-h)-(\operatorname{orb}x)(t)\|\le\|T(t-h)\|_{\mathfrak L(E)}\|x-T(h)x\|_E\tag7$$ for all $h\in[0,t]$. Now I don't see how we can conclude, since there is generally no bound for $\|T(t-h)\|_{\mathfrak L(E)}$.

REPLY [0 votes]: The continuity at $s=0$ implies the continuity on $[t,t+h]$ (by the semigroup property and the exponential bound). Hence, the Bochner integrability.
Edit:
For $t\ge h\ge 0$:
\begin{align*}
\|(\operatorname{orb}x)(t-h)-(\operatorname{orb}x)(t)\| &\le\|T(t-h)\|_{\mathfrak L(E)}\|x-T(h)x\|_E \\
& \le M e^{\omega (t-h)} \|x-T(h)x\|_E \; \to 0 \text{ as } h \to 0^+.
\end{align*}


text:
\begin{document}

\maketitle

\begin{abstract}
Given a measurable space $(X, \mathcal{M})$ there is a (Galois) connection between sub-$\sigma$-algebras of $\mathcal{M}$ and equivalence relations on $X$. On the other hand equivalence relations on $X$ are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-$\sigma$-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.
\end{abstract}

\section{Introduction}

Recently, measurable spaces, and in particular analytic spaces, have been proposed for modelling probabilistic systems \cite{deshetalbisim,eedcongbisim,dadelapa}. The continuous, uncountable nature of such systems is reflected by the uncountable state spaces provided by the measurable setting. Many notions of equivalence or similarity of probabilistic systems, such as behavioural equivalence, logical equivalence or bisimulation, can be modelled in this setting and can then be used for system simplification and system verification. One of the tools in modelling equivalences and bisimilarity in the setting of analytic spaces is the notion of a congruence on a stochastic relation \cite{eed1,eedstochrel}. Such a congruence is given by a pair of smooth equivalence relations on the underlying analytic spaces satisfying certain conditions.

In recent work, Doberkat has investigated the set of smooth equivalence relations of an analytic space on order- and lattice-theoretic properties. Results include an example showing that congruences are not closed under intersection in the lattice of all equivalence relations \cite{eedconglatt} and a construction having similar properties to a complement or negation \cite{eednegcong}. In the end of this second article he asks for a domain-theoretic investigation of the set of smooth equivalence relation on an analytic space, more specifically for a characterisation of the way-below order on posets of smooth relations.

The way-below order comes from domain theory \cite{abjudomain}, probably the most popular framework for denotational semantics of programming languages. The key idea of domain theory is to equip a set denoting a datatype with an order which reflects the amount of information that the respective data carries. The way-below order is derived from this original order to model uniform approximation of data elements and programs. Thus, a well-behaved way-below order on smooth equivalence relations may open the door towards a theory of approximations and computability of certain kind of similarities.

In this paper we attempt a domain-theoretic investigation of posets of sub-$\sigma$-algebras. During this investigation, we generalise one of Doberkat's results in \cite{eedconglatt} and show that the posets of smooth equivalence relations on an analytic space ordered by inclusion and reverse inclusion might have trivial orders of approximation, hence they do not form pleasant domains in general. To obtain this result, we will use the isomorphism between smooth equivalence relations on the analytic space $(X, \mathcal{M})$ and countably-generated sub-$\sigma$-algebras of $\mathcal{M}$, as defined in section 1.7 of \cite{eed2}. This isomorphism is induced by a Galois connection between the set of all equivalence relations on $X$ and sub-$\sigma$-algebras of $\mathcal{M}$, as mentioned in \cite{dadelapa}. We also use results of \cite{raorao}, where the general lattice-theoretic properties of the set of sub-$\sigma$-algebras of a given $\sigma$-algebra $\mathcal{M}$ are examined.

The paper is organised as follows. In section 2, we fix our measure theoretic and domain theoretic notation and recall some basic results, in particular the Galois connection between sub-$\sigma$-algebras and equivalence relations. In section 3, we investigate posets of sub-$\sigma$-algebras of a given $\sigma$-algebra $\mathcal{M}$ with respect to the set-inclusion order. Along the way we show that the countably-generated sub-$\sigma$-algebras of an analytic space do in general not form directed complete posets and use this result to prove that the category of analytic spaces does not have pushouts. Finally, we show that with respect to the inclusion order, various posets of sub-$\sigma$-algebras of the usual Borel-$\sigma$-algebra of the real numbers have a trivial approximation order. In section 4, we investigate the opposite of the inclusion order on the posets of sub-$\sigma$-algebras, which arguably is the more interesting order from the category-theoretical viewpoint. Under the Galois connection it corresponds directly to the inclusion order on the equivalence relations. We show that also with respect to this order the posets of sub-$\sigma$-algebras are not directed complete and might have a trivial way-below order. Again we show that this in particular holds for the standard Borel $\sigma$-algebra on the real numbers and hence the set of smooth relations on this measurable space has a trivial order of approximation when equipped with the usual inclusion order. We finish the paper with some final remarks in section 5.

\section{Preliminaries}

Let us start by introducing our notation and giving some basic results in this section. First we give the measure-theoretic background, followed by equivalence relations on measurable spaces and the Galois connection between them and sub-$\sigma$-algebras, and finally the domain-theoretic counterpart.

\subsection{Measurable spaces}

We do not give details or motivations about the basic notions in this section, for a more detailed treatment the reader is referred to \cite{eed1,srivastava}.

\begin{itemize}
\item A \emph{measurable space} $(X, \mathcal{M})$ is a set $X$ equipped with a $\sigma$-algebra $\mathcal{M}$ of subsets of $X$. The elements of $\mathcal{M}$ are called \emph{measurable sets}. A map $f: (X, \mathcal{M}) \rightarrow (Y, \mathcal{N})$ between measurable spaces is called a \emph{measurable map} if for all $A \in \mathcal{N}$, it holds that $f^{-1} (A) \in \mathcal{M}$.

\item If $\mathcal{M}$ and $\mathcal{N}$ are $\sigma$-algebras on a set $X$ with $\mathcal{N} \subseteq \mathcal{M}$, then $\mathcal{N}$ is a \emph{sub-$\sigma$-algebra} of $\mathcal{M}$.

\item A $\sigma$-algebra $\mathcal{M}$ on $X$ is called \emph{countably-generated} if there exists a sequence $(A_n)_{n \in \mathbb{N}}$ of subsets of $X$ such that $\mathcal{M} = \sigma (\{A_n\}_{n \in \mathbb{N}})$, i.e. the smallest $\Sigma$-algebra containing all the $A_n$.

\item A measurable space $(X, \mathcal{M})$ is called a \emph{Standard Borel space} if there exists topology $\mathcal{T}$ on $X$ such that $(X, \mathcal{T})$ is a Polish space \cite{kuratowski} and $\mathcal{M} = \sigma (\mathcal{T})$, i.e. the smallest $\sigma$-algebra containing $\mathcal{T}$.

\item An \emph{analytic space} $(X, \mathcal{M})$ is a measurable space which is isomorphic to the image of some measurable map $f: (Y, \mathcal{N}) \rightarrow (Y', \mathcal{N}')$ between Standard Borel spaces \cite{kechriscdst}.
\end{itemize}

The topological origin of analytic and Standard Borel spaces yields the machinery to show many interesting results and hence makes them an ideal setting to study continuous probabilistic systems. Some of these results can be extended to the class of all measurable spaces, as done by Schubert in \cite{schubcoalgmodlog}. In this generalisation the following properties, which are always satisfied in analytic spaces, play important r\^{o}les.

\begin{itemize}
\item A measurable space $(X, \mathcal{M})$ is called \emph{separated} if the elements of $\mathcal{M}$ separate the points of $X$, i.e. if for any distinct $x,y \in X$ there exists some $A \in \mathcal{M}$ containing $x$ but not $y$. A measurable space $(X, \mathcal{M})$ is called \emph{separable} if it is separated and $\mathcal{M}$ is countably-generated.
\end{itemize}\label{ref0}

In this paper we are interested in studying the properties of partially ordered sets (henceforth \emph{posets}) of sub-$\sigma$-algebras of measurable and analytic spaces. Notice that for any set $X$ the power set $\mathcal{P}(X)$ forms the largest and $\{\emptyset, X\}$ the smallest $\sigma$-algebra on $X$. Moreover if $\{\mathcal{A}_i\}_{i \in I}$ is a family of $\sigma$-algebras on $X$, then $\bigcap_{i \in I} \mathcal{A}_i$ is also a $\sigma$-algebra on $X$, and it is the largest common sub-$\sigma$-algebra of all the $\mathcal{A}_i$. It follows that the $\sigma$-algebras on a set $X$ form a complete lattice under the usual set-inclusion order. The meet is given by intersection and the join is given as $\bigvee_{i \in I} \mathcal{A}_i = \sigma(\bigcup_{i \in I} \mathcal{A}_i)$.

For a given measurable space $(X, \mathcal{M})$ we fix the following notation.

\begin{itemize}
\item By $\mathrm{Sub}_\sigma(\mathcal{M})$ we denote the set of all sub-$\sigma$-algebras of $\mathcal{M}$. The inclusion order $\subseteq$ makes $(\mathrm{Sub}_\sigma(\mathcal{M}), \subseteq)$ into a complete lattice. We write $\sqsubseteq$ for the opposite order (i.e. $\mathcal{A} \sqsubseteq \mathcal{A}'$ if $\mathcal{A} \supseteq \mathcal{A}'$) on these sets, and then also $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ is a complete lattice.
\end{itemize}

In the study of analytic spaces, the countably generated $\sigma$-algebras are of particular interest \cite{eed1,eed2}, motivating the following notation.

\begin{itemize}
\item By $\mathrm{Sub}^{\omega}_\sigma(\mathcal{M})$ we denote the set of countably-generated sub-$\sigma$-algebras of $\mathcal{M}$. With the inherited orders, $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \subseteq)$ and $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \sqsubseteq)$ are posets.
\end{itemize}

Notice that the set of all $\sigma$-algebras on a set $X$ is given by $\mathrm{Sub}_\sigma(\mathcal{P}(X))$, and the set of all countably-generated $\sigma$-algebras by $\mathrm{Sub}^{\omega}_\sigma(\mathcal{P}(X))$. Thus our results below can also be adjusted to a setting where no $\sigma$-algebra of measurable sets is assumed.

Let us make the following trivial observation.

\begin{prop}
Let $(X, \mathcal{M})$ be a measurable space. Then $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \subseteq)$ is a $\vee$-semilattice and has countable suprema, accordingly $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \sqsubseteq)$ is a $\wedge$-semilattice and has countable infima.\label{refx3}
\end{prop}

Below we will show that in general they do not support the corresponding opposite operation.

\subsection{Equivalence relations}

As we have mentioned in the introduction, there is a close connection between equivalence relations and sub-$\sigma$-algebras of a measurable space $(X, \mathcal{M})$. This connection is made precise by the following result, the proof of which is straightforward.

\begin{lemm}
Let $(X, \mathcal{M})$ be a measurable space and $\sim \subseteq X \times X$ be an equivalence relation. Then the set $\mathcal{M}_\sim$ of $\sim$-invariant sets of $\mathcal{M}$ forms a sub-$\sigma$-algebra of $\mathcal{M}$.
\end{lemm}

Notice that in general not all $\sigma$-algebras are defined like this. For instance consider $\cC$, the $\sigma$-algebra of countable and cocountable subsets of a measurable space $(X, \mathcal{P}(X))$, where $X$ is some uncountable set. Since $\cC$ contains all the singletons, the only equivalence relation which could possibly induce it according to the lemma above is equality. However every subset of $X$ is $=$-invariant, and so we have $\mathcal{P}(X)_= = \mathcal{P}(X) \neq \cC$. Thus, ingeneral, the set of $\sigma$-algebras of invariant sets of equivalence relations forms a proper subset of the set of all sub-$\sigma$-algebras of a measurable space $(X, \mathcal{M})$.

\begin{defi}
Let $(X, \mathcal{M})$ be a measurable space. We denote the set of equivalence relations on $X$ by $\mathrm{Eq} (X)$. For an element $\sim \in \mathrm{Eq} (X)$, the \emph{$\sim$-induced sub-$\sigma$-algebra} $\mathcal{M}_\sim$ of $\mathcal{M}$ is given by the $\sim$-invariant sets of $\mathcal{M}$. We denote the subset of $\Sub (\mathcal{M})$ given by equivalence relation induced sub-$\sigma$-algebras by $\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M})$.\label{x0}
\end{defi}

Notice that there is a converse to the construction of $\sigma$-algebras of invariant sets.

\begin{defi}
For a given $\sigma$-algebra $\mathcal{M}$ on a set $X$, we denote by $\sim_{\mathcal{M}}$ the \emph{equivalence relation induced by $\mathcal{M}$}, given by
$$x \: \sim_\mathcal{M} \: y := (\forall A \in \mathcal{M}. \: x \in A \ifof y \in A).$$\label{x1}
\end{defi}

It is well-known that for such equivalence relations it suffices to consider the behaviour on a generator of the given $\sigma$-algebra, as the following result shows, whose proof can be found as Lemma 3.1.6 in \cite{srivastava}.

\begin{lemm}
Let $\cG$ be a generator of a $\sigma$-algebra $\mathcal{M}$ on $X$. Then $x \: \sim_\mathcal{M} \: y$ if and only if for all $G \in \cG$ it holds that:
$$x \in G \; \ifof \; y \in G.$$
\end{lemm}

Of particular interest in the study of analytic spaces have been those equivalence relations which are induced by a countably-generated $\sigma$-algebra.

\begin{defi}
A \emph{smooth relation} on a measurable space $(X, \mathcal{M})$ is an equivalence relation induced by a countably generated sub-$\sigma$-algebra of $\mathcal{M}$. We denote the subset of smooth relations of $\mathrm{Eq}(X)$ by $\mathrm{Sm}(X, \mathcal{M})$.
\end{defi}

Smooth relations are of paramount interest in the study of analytic spaces because they characterise exactly those equivalence relations for which the quotient space is again analytic as the following pivotal result shows. It is shown as Lemma 1.52 and Proposition 1.53 in \cite{eed1}.

\begin{theo}
Let $(X, \mathcal{M})$ be an analytic space and $\sim$ be an equivalence relation on $X$. Then the following are equivalent:
\begin{itemize}
 \item[(I)] $\sim$ is smooth,

 \item[(II)] $\sim$ is the kernel relation for a measurable map $f: (X, \mathcal{M}) \rightarrow (Y, \mathcal{N})$ into an analytic space $(Y, \mathcal{N})$,

 \item[(III)] the quotient space $(X / \! \sim, \mathcal{M}_\sim)$ is analytic.
\end{itemize}\label{x3}
\end{theo}

\begin{coro}
For an analytic space $(X, \mathcal{M})$ the sets $\mathrm{Sub}^{\omega}_\sigma(\mathcal{M})$ and $\mathrm{Sm}(X, \mathcal{M})$ are isomorphic.\label{x2}
\end{coro}

This isomorphism can be extended along intrinsic orders on these sets. In fact, it is the restriction of a more general order theoretic connection between equivalence relations and sub-$\sigma$-algebras on measurable spaces, namely a Galois connection.

\begin{defi}
A \emph{Galois connection} (or \emph{adjunction}) between posets $(X, \leq)$ and $(Y, \sqsubseteq)$ is a pair of monotone maps $f: (X, \leq) \leftrightarrow (Y, \sqsubseteq) :g$ for which it holds that $f(x) \sqsubseteq y$ if and only if $x \leq g(y)$. In this case $f$ is called the \emph{lower adjoint} and $g$ the \emph{upper adjoint}.
\end{defi}

For the remainder of this part, let $(X, \mathcal{M})$ be some fixed measurable space. To avoid confusion, we write $\lessdot$ for the inclusion order on $\Eq (X)$, i.e. $\sim \lessdot \smile$ if $x \sim y$ implies $x \smile y$ for all $x,y \in X$. Then $(\Eq (X), \lessdot)$ is a complete lattice where meet is defined by intersection of the given relations in $X \times X$ and join by the equivalence relation generated by the union of the given relations in $X \times X$. It is essentially shown in \cite{dadelapa} that the constructions defined in Definitions \ref{x0} and \ref{x1} form a Galois connection between $(\mathrm{Eq}(X), \lessdot)$ and $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$.

\begin{lemm}
For elements $\mathcal{A}, \cB \in \mathrm{Sub}_\sigma(\mathcal{M})$ with $\mathcal{A} \sqsubseteq \cB$, it holds that $\sim_\mathcal{A} \lessdot \sim_\cB$, and conversely for elements $\sim, \smile \in \mathrm{Eq}(\mathcal{M})$ with $\sim \lessdot \smile$, it holds that $\mathcal{A}_\sim \sqsubseteq \mathcal{A}_\smile$. The maps $\mathcal{M}_{(-)} : (\mathrm{Eq}(X), \lessdot)  \leftrightarrow (\Sub (\mathcal{M}), \sqsubseteq) : \sim_{(-)}$ form a Galois connection, with $\mathcal{M}_{(-)}$ being the lower and $\sim_{(-)}$ being the upper adjoint.
\end{lemm}

Notice that in general we do not have an isomorphism between $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ and $(\mathrm{Eq}(X), \lessdot)$. The example of the countable-cocountable $\sigma$-algebra above shows that $\mathcal{M}_{(-)}$ is not surjective, and the following consideration, which is Example 4.12 in \cite{dadelapa}, shows that $\sim_{(-)}$ cannot be surjective. Let $T$ be a non-measurable subset of $(\mathbb{R}, \cB)$ (with the usual Euclidean Borel algebra) and define $\sim$ to have equivalence classes $T$ and $\mathbb{R} \setminus T$. Then $\sim$ cannot be induced by any sub-$\sigma$-algebra of $\cB$ as this would have to contain $T$.

Notice that we can characterise the image of one of the two compositions.

\begin{lemm}
The image of $\mathcal{M}_{\sim_{(-)}}$ is given by $\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M})$ which therefore is the set of fixed points of $\mathcal{M}_{\sim_{(-)}}$. Thus $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}), \sqsubseteq)$ is a complete lattice with join defined as:
$$\bigvee \mathcal{A}_i := \mathcal{M}_{\bigvee \sim_{\mathcal{A}_i}} \hspace{1cm}.$$\label{ref4}
\end{lemm}

The other composition is the more problematic one, and we do not know how to characterise it. Nevertheless we mention Corollary 4.10 of \cite{dadelapa}.

\begin{prop}
Let $(X, \mathcal{M})$ be a measurable space and $\sim$ be an equivalence relation on $X$ such that all equivalence classes are in $\mathcal{M}$. Then $\sim = \sim_{\mathcal{M}_\sim}$.\label{ref3}
\end{prop}

Notice that this is the case for smooth equivalence relations, because we have:

\begin{prop}
For a smooth equivalence relation on a measurable space $(X, \mathcal{M})$ all equivalence classes are measurable sets.\label{x4}
\end{prop}

\begin{prove}
Let $x \in X$ and $(A_n)_{n \in \mathbb{N}}$ be a countable generator inducing the equivalence relation $\sim$. Then it holds that $[x]_\sim = \bigcap \{ A_n |\: x \in A_n\} \cap \bigcap \{ X \setminus A_n |\: x \notin A_n\}$, which is a measurable set.\qed
\end{prove}

Thus, we get that smooth equivalence relations are fixed points of $\sim_{\mathcal{M}_{(-)}}$. As mentioned, on analytic spaces the isomorphism of Corollary \ref{x2} becomes an order isomorphism between $(\Sm (X, \mathcal{M}), \lessdot)$ and $(\mathrm{Sub}^{\omega}_\sigma (\mathcal{M}), \sqsubseteq)$, see e.g. section 1.7 of \cite{eed2}.

\begin{lemm}
On an analytic space $(X, \mathcal{M})$, the Galois connection $(\Sub (\mathcal{M}), \sqsubseteq) \leftrightarrow (\mathrm{Eq}(X), \lessdot)$ restricts to an isomorphism $(\mathrm{Sub}^{\omega}_\sigma (\mathcal{M}), \sqsubseteq) \cong (\mathrm{Sm}(X, \mathcal{M}), \lessdot)$.\label{ref2}
\end{lemm}

Observe, that this result also shows that for an analytic space $(X, \mathcal{M})$ we have $\mathrm{Sub}^{\omega}_\sigma (\mathcal{M}) \subseteq \mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}) \subseteq \Sub (\mathcal{M})$. Both inclusions are proper in general. For the last one, this is clear, as the example of the countable-cocountable sub-$\sigma$-algebra on the real numbers has shown. The first one can be obtained by using the counterexample of Section 3.2 in \cite{eedconglatt}. Once we leave the realm of analytic spaces, the first inclusion need not be valid: The usual Euclidean Borel-$\sigma$-algebra on the real numbers is countably generated but not in $\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{P}(\mathbb{R}))$.

\subsection{Domain Theory}

Also in this part we refrain from giving a motivation or detailed background and simply introduce the concepts we need below. For more details on domain theory we suggest the introductory survey \cite{abjudomain}.

\begin{itemize}
\item Let $(X,\leq)$ be a partially ordered set. A subset $D \subseteq X$ is \emph{directed} if for any $x,y \in D$ there exists some $z \in D$ with $x,y \leq z$. The poset $(X, \leq)$ is called \emph{directed complete} if every directed subset of $X$ has a supremum $\dsup D$ in $X$.

\item In a poset $(X, \leq)$ an element $x$ is \emph{way-below} another element $y$, written as $x \ll y$, if for any directed $D \subseteq X$ which has a supremum $\dsup D$ with $y \leq \dsup D$, there exists some $d \in D$ with $x \leq d$.
\end{itemize}

The way-below relation is of paramount interest in domain theory as it provides a meaning of uniform approximation of elements. It is sometimes called the \emph{order of approximation}.

In fact, directed completeness is a generalisation of chain completeness which we will also use below.

\begin{itemize}
\item Let $(X,\leq)$ be a partially ordered set. By an \emph{$\w$-chain} we refer to a sequence $\{x_n\}_{n \in \mathbb{N}}$ which is increasing in $X$, i.e. for all $n \in \mathbb{N}$, it holds that $x_n \leq x_{n+1}$. The poset $(X, \leq)$ is called \emph{$\w$-chain complete} if every $\w$-chain in $X$ has a supremum $\dsup_n x_n$ in $X$.

\item In a poset $(X, \leq)$ an element $x$ is \emph{$\w$-way-below} another element $y$, written as $x \ll_\w y$, if for any $\w$-chain $\{x_n\}_{n \in \mathbb{N}}$ in $X$ which has a supremum $\dsup_n x_n$ with $y \leq \dsup_n x_n$, there exists some $n_0 \in \mathbb{N}$ with $x \leq x_{n_0}$.
\end{itemize}

These definitions should suffice to follow the technical development in this paper. On a detailed analysis on how the directed and $\w$-chain notions relate we refer the reader to section 2.2.4 of \cite{abjudomain}.

\section{The inclusion order}

In this section we investigate the lattices $(\mathrm{Sub}_\sigma(\mathcal{M}), \subseteq)$ and $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}), \subseteq)$ and also the poset $(\Sub^{\w} (\mathcal{M}), \subseteq)$ on their properties. Let us start by recalling some more or less obvious results.

\begin{prop}
For any measurable space $(X, \mathcal{M})$, every element $\mathcal{A} \in \Sub (\mathcal{M})$ can be obtained as the directed supremum of finitely-generated sub-$\sigma$-algebras of $\mathcal{M}$.
\end{prop}

Notice that (surprisingly) the finitely-generated sub-$\sigma$-algebras of $\mathcal{M}$ need not be compact elements in the lattice: Consider a point $x \in \mathbb{R}$ and the $\sigma$-algebras $\mathcal{A}_n := \sigma (\{U_{\frac{1}{k}} (x) |\: k \leq n \})$, where $U_{\frac{1}{k}} (x) := \{y \in X | \: |x-y| < \frac{1}{k} \}$ denotes the open ball around $x$ with radius $\frac{1}{k}$. Then $\{x\} \in \dsup \mathcal{A}_n$ but $\{x \} \notin \mathcal{A}_n$ for any $n \in \mathbb{N}$.

Below we will show below that the way-below order on $(\mathrm{Sub}_\sigma(\cB), \subseteq)$, where $\cB$ is the usual Borel $\sigma$-algebra on $\mathbb{R}$ is trivial. But let us first mention some more general results.

\begin{coro}
Suppose $(\Sub^{\w} (\mathcal{M}), \subseteq)$ is directed complete. Then it is a monotone retract of $(\Sub (\mathcal{M}), \subseteq)$ with section-retraction pair $\iota: (\Sub^{\w} (\mathcal{M}), \subseteq) \leftrightarrow (\Sub (\mathcal{M}), \subseteq): \kappa$ such that $\id_{\Sub (\mathcal{M})} \leq \iota \circ \kappa$. It follows that in this case $\mathcal{M}$ is countably-generated.
\end{coro}

Notice that we do not know whether $\Sub^{\w} (\mathcal{M})$ forms a continuous retract of $\Sub (\mathcal{M})$ if it is directed complete, since the inclusion map $\iota$ might not preserve directed suprema. In fact it will do so only if $\mathrm{Sub}^{\omega}_\sigma (\mathcal{M}) \equiv \Sub (\mathcal{M})$.

Recall a counterexample in section 3.2 of \cite{eedconglatt} which shows that $\Sub^{\w} (\mathcal{M})$ is in general not closed under the intersection operation $\cap$ in $\Sub (\mathcal{M})$ even for analytic spaces.

\begin{prop}
In general, for an analytic space $(X, \mathcal{M})$ the poset $(\Sub^{\w} (\mathcal{M}), \subseteq)$ is not a lattice.
\end{prop}

\begin{prove}
Consider the mentioned example which constructs countably-generated sub-$\sigma$-algebras $\mathcal{A}, \cB$ of an analytic space $(X, \mathcal{M})$, for which $\mathcal{A} \cap \cB$ is not countably-generated. Suppose that nevertheless $(\Sub^{\w} (\mathcal{M}), \subseteq)$ has a meet operation $\wedge^\w$. Then, since $\mathcal{A} \wedge^\w \cB \subseteq \mathcal{A}, \cB$, it holds that $\mathcal{A} \wedge^\w \cB \subseteq \mathcal{A} \cap \cB$ in $\Sub (\mathcal{M})$. Moreover, for each $A \in \mathcal{A} \cap \cB$, it holds that $\sigma (\{A\}) \in \Sub^{\w} (\mathcal{M})$ and $\sigma (\{A\}) \subseteq \mathcal{A}, \cB$. Thus we get that $A \in \sigma (\{A\}) \subseteq \mathcal{A} \wedge^\w \cB$, showing $\mathcal{A} \cap \cB \subseteq \mathcal{A} \wedge^\w \cB$, and we conclude that they must be equal, which contradicts the counterexample.\qed
\end{prove}

This implies the following result, which generalises Doberkat's Corollary 3.19 in \cite{eedconglatt} which shows that analytic spaces are not closed under pushouts in the category of measurable spaces and measurable maps.

\begin{theo}
The category of analytic spaces and measurable maps does not have all pushouts.
\end{theo}

Now we show that the way-below relation on the sub-$\sigma$-algebras $(\Sub (\cB), \subseteq)$ of the standard Borel $\sigma$-algebra $\cB$ on the real numbers $\mathbb{R}$ is trivial.

\begin{theo}
In $(\Sub (\cB), \subseteq)$, it holds that $\mathcal{A} \ll \cB$ if and only if $\mathcal{A} = \{\emptyset, \mathbb{R} \}$.
\end{theo}

\begin{prsk}
Suppose $\{\emptyset, \mathbb{R} \} \neq \mathcal{A}$ and fix a Borel set $B \subseteq \mathbb{R}$ in $\mathcal{A}$ such that $\emptyset \neq B \neq \mathbb{R}$. Pick some point $y \in \mathbb{R}$ of the topological boundary of $\mathcal{A}$ and modify the sets of the generator $\cO := \{ U_q (s) |\: r,s \in \Q \}$ of $\cB$ as follows. Enumerate the sets, e.g. $\{V_n\}_{n \in \mathbb{N}}$, and either add a small ball $U_{\frac{1}{n}} (y)$ to $V_n$ or remove it according as to whether $y$ is a member of $V_n$ or not. The modified sets still generate $\cB$ hence one can obtain $\cB$ as the directed supremum of the $\sigma$-algebras generated by finitely many such sets. However, the modification ensures that none of these finite $\sigma$-algebras contains $B$ and hence cannot have $\mathcal{A}$ as a subset.\qed
\end{prsk}

\begin{coro}
For the way-below relation $\ll$, resp. the $\w$-chain way-below relation $\ll_{\w}$, on $(\Sub (\cB), \subseteq)$, $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\cB), \subseteq)$ and $(\mathrm{Sub}^{\omega}_\sigma (\cB), \subseteq)$, it holds that $\mathcal{A} \ll \cB$ (resp. $\mathcal{A} \ll_{\w} \cB$) if and only if $\mathcal{A} = \{\emptyset, \mathbb{R}\}$.
\end{coro}

This shows that from a domain-theoretical viewpoint, the inclusion order $\subseteq$ does in general not yield pleasant posets of sub-$\sigma$-algebras, even in the restricted setting of analytic spaces or Standard Borel spaces.

\section{The opposite order}

After our investigation of the inclusion order in the previous section, we now investigate the opposite order on posets of sub-$\sigma$-algebras. Observe that from a category-theoretical viewpoint this order is the more interesting one, since whenever $\mathcal{A} \sqsubseteq \cB$ for $\mathcal{A},\cB \in \Sub (\mathcal{M})$ for a measurable space $(X, \mathcal{M})$, the identity map $(X, \mathcal{A}) \rightarrow (X, \cB)$ is measurable. Similarly, if $\mathcal{A} \sqsubseteq \cB$ for $\mathcal{A}, \cB \in \mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M})$ then the measurable quotient map $X \rightarrow X /\! \sim_{\cB}$ factors through the quotient $X \rightarrow X /\! \sim_{\mathcal{A}}$.

We start our investigation by looking at the atoms in the lattice $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ for some fixed measurable space $(X, \mathcal{M})$. These atoms have been identified in \S 18 of \cite{raorao}, and we describe their construction in the following. The basic idea is to lift equivalence relations on a measurable space $(X, \mathcal{M})$ to the set of probability measures $\cS(\mathcal{M})$. Recall that a $0-1$-measure on $\mathcal{M}$ is a probability measure which only takes the values $0$ and $1$. For instance every point measure $\delta_x$, defined by $\delta_x (A) = 1$ if $x \in A$ and $\delta_x (A) = 0$ otherwise, is a $0-1$-measure.

\begin{prop}
For any family $\{ S_i \}_{i \in I}$ of pairwise disjoint sets of $0-1$-measures the set of $\{S_i\}_{i \in I}$-invariant measurable sets, defined as:
$$\mathcal{M}^{\langle S_i\rangle_{i \in I}} := \{ A \in \mathcal{M} |\: \forall i \in I.\forall \mu, \mu' \in S_i. \: \mu (A) = \mu'(A) \}$$
is a sub-$\sigma$-algebra of $\mathcal{M}$.\label{refx1}
\end{prop}

\begin{prove}
Straightforward using the $\pi-\lambda$-Theorem \cite{eed1}.\qed
\end{prove}

Rao and Rao \cite{raorao} have shown that the atoms in $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ are precisely the sub-$\sigma$-algebras of the form $\mathcal{M}^S$ for set $S$ containing two distinct $0-1$-measures.

\begin{lemm}
Let $(X, \mathcal{M})$ be a measurable space and $\mu,\mu'$ be a pair of distinct $0-1$-measures. Then $\mathcal{M}^{\{\mu,\mu'\}}$ is an atom in $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$. Conversely every atom of $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ is of the form $\mathcal{M}^{\{\mu,\mu'\}}$.\label{x5}
\end{lemm}

So let us take a closer look at $0-1$-measures.

\begin{prop}
Let $\mu$ be a $0-1$-measure on a measurable space $(X, \mathcal{M})$. If the set $\bigcap \{A \in \mathcal{M} |\: \mu(A) =1\}$ is nonempty, then it cannot be separated by $\mathcal{M}$. In particular if $(X, \mathcal{M})$ is separated then a $0-1$-measure $\mu$ is either a point measure or $\bigcap \{A \in \mathcal{M} |\: \mu(A) =1\} = \emptyset$.
\end{prop}

\begin{prove}
Suppose otherwise $x,y \in \bigcap \{A \in \mathcal{M} |\: \mu(A) =1\}$ could be separated by $B$, i.e. $x \in B$ and $y \notin B$. Then we get $1 = \mu (X) = \mu(B) + \mu(X \setminus B) = 0$ which yields an obvious contradiction.\qed
\end{prove}

Notice that it is possible for the intersection to be empty. For instance in $(\mathbb{R}, \cC)$, $\cC$ being the countable-cocountable $\sigma$-algebra on $\mathbb{R}$, there is a $0-1$-measure which gives all cocountable sets measure $1$. This counterexample appears amongst others in \cite{raorao}. However, if we restrict ourselves to countably-generated measurable spaces we get the following result which is essentially shown in \S 4 of \emph{op.cit.}.

\begin{lemm}
If $(X,\mathcal{M})$ be a countably-generated measurable space. Then any $0-1$-measure $\mu$ is a point measure.\label{refx2}
\end{lemm}

\begin{coro}
If $(X,\mathcal{M})$ is a separable measurable space then any $0-1$-measure $\mu$ is a unique point measure.\label{ref6b}
\end{coro}

The case of the countable-cocountable $\sigma$-algebra $\cC$ on $\mathbb{R}$ shows that these results do not transfer to sub-$\sigma$-algebras of separable measurable spaces. It also shows another thing, namely that in general the lattices $(\Sub (\mathcal{M}), \sqsubseteq)$ are not atomic, because as a sub-$\sigma$-algebra of the standard Borel sets $\cB$, $\cC$ is not contained in any $\cB^{\{\mu,\mu'\}}$ for a pair of $0-1$-measures.

Surprisingly, this is different in the lattice $(\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M}), \sqsubseteq)$.

\begin{prop}
For every measurable space $(X,\mathcal{M})$, the lattice $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}), \sqsubseteq)$ is atomic.
\end{prop}

\begin{prsk}
In terms of equivalence relations, the atoms of $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}), \sqsubseteq)$ correspond exactly to the relations $\sim^{\{a,b\}}$ which identify elements $a,b \in X$ which can be separated by $\mathcal{M}$. It is not hard to see that every equivalence relation induced by a sub-$\sigma$-algebra of $\mathcal{M}$ is obtained as the supremum of such relations below it, and that this carries over through the Galois connection.\qed
\end{prsk}

Let us now turn towards smooth equivalence relations and countably-generated $\sigma$-algebras. In the following we define the notion of finitely generated partitions on $(X, \mathcal{M})$, show that these are closed under suprema in $(\mathrm{Eq}(X), \lessdot)$ and that they are smooth relations in the case of analytic spaces. We then use these results to show that in general $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \sqsubseteq)$ is not a dcpo and that the way-below order on it may be trivial.

\begin{defi}
Let $(X, \mathcal{M})$ be a measurable space, and $\{S_f\}_{f \in F}$ be a finite family of pairwise disjoint, finite subsets of $X$. We define a relation $\sim^{\langle S_f\rangle_{f \in F}}$ as the equivalence relation on $X$ which has the equivalence classes $S_f$ for $f \in F$ and $\{x\}$ for $x \in X \setminus \bigcup_{f \in F} S_f$, i.e. $\sim^{\langle S_f\rangle_{f \in F}}$ identifies the elements of the $S_f$. We call such an equivalence relation a \emph{finitely generated partition} of $X$.

For a finitely generated partition  $\{S_f\}_{f \in F}$ of $X$, we call the sub-$\sigma$-algebra of $\sim^{\langle S_f\rangle_{f \in F}}$-invariant subsets of $\mathcal{M}$ an \emph{fgp-sub-$\sigma$-algebra} of $\mathcal{M}$.
\end{defi}

Of course, the fgp-sub-$\sigma$-algebras of $\mathcal{M}$ are precisely the images of the finitely-generated partitions under the Galois connection $(\Sub (\mathcal{M}),\sqsubseteq) \leftrightarrow (\mathrm{Eq}(X), \lessdot)$. In fact, they correspond to the finite suprema of the atoms of $\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M})$, as the following result shows whose proof is straightforward.

\begin{lemm}
Let $X$ be a set. In $(\mathrm{Eq}(X),\lessdot)$ a finite join of finitely-generated partitions is again a finitely generated partition.\label{ref6}
\end{lemm}

\begin{rem}
In fact from the domain-theoretical viewpoint the finitely-generated partitions are of particular interest in $(\mathrm{Eq}(X), \lessdot)$, because the are exactly the compact elements making this lattice algebraic.
\end{rem}

The following result is now a straightforward consequence of the considerations we just made.

\begin{coro}
In a measurable space $(X, \mathcal{M})$, every element of $(\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M}), \sqsubseteq)$ is the directed supremum of the fgp-sub-$\sigma$-algebras of $\mathcal{M}$ that are below it.
\end{coro}

As we have seen in section 2.2, for an analytic space $(X, \mathcal{M})$ the set $\mathrm{Sub}^{\omega}_\sigma(\mathcal{M})$ is a subset of $\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M})$. We now show that it includes all fgp-sub-$\sigma$-algebras of $\mathcal{M}$.

\begin{prop}
Let $(X, \mathcal{M})$ be an analytic space. Then any fgp-sub-$\sigma$-algebra $\mathcal{A}$ of $\mathcal{M}$ is countably-generated. Equivalently, finitely generated partitions on $X$ are smooth equivalence relations on $(X, \mathcal{M})$.\label{ref7}
\end{prop}

\begin{prsk}
One shows that a finitely-generated partition $\sim^{\langle S_f\rangle_{f \in F}}$ is smooth on $(X, \mathcal{M})$ by applying Theorem \ref{x3} to the kernel of a measurable map which picks one representative for each equivalence class.\qed
\end{prsk}

So let us summarise.

\begin{coro}
For every analytic space $(X, \mathcal{M})$ the poset $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \sqsubseteq)$ is atomic, i.e. every element can be obtained as the supremum of the atoms below it. Moreover, $(\mathrm{Sub}^{\omega}_\sigma(\mathcal{M}), \sqsubseteq)$ contains all the atoms of $(\mathrm{Sub}_\sigma(\mathcal{M}), \sqsubseteq)$ (and $(\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M}), \sqsubseteq)$) and every element of $\mathrm{Sub}^{\mathrm{eq}}_\sigma(\mathcal{M})$ can be obtained as directed supremum of elements of $\mathrm{Sub}^{\omega}_\sigma(\mathcal{M})$. In particular, every smooth equivalence relation on $(X, \mathcal{M})$ is the directed join of the finitely generated partitions included in it.
\end{coro}

With the counterexample of section 3.2 in \cite{eedconglatt} we can show again:

\begin{prop}
For an analytic space $\mathrm{Sub}^{\omega}_\sigma(\mathcal{M})$ is not directed complete in general.\label{refx5}
\end{prop}

Furthermore we can show again that for the usual Borel-$\sigma$-algebra $\cB$ on $\mathbb{R}$ the way-below relation on $(\mathrm{Sub}^{\mathrm{eq}}_\sigma(\cB), \sqsubseteq)$ and $(\mathrm{Sub}^{\omega}_\sigma(\cB), \sqsubseteq)$ is trivial.

\begin{theo}
On $(\mathrm{Sub}^{\mathrm{eq}}_\sigma(\cB), \sqsubseteq)$ and $(\mathrm{Sub}^{\omega}_\sigma(\cB), \sqsubseteq)$ the way-below relation $\ll$ is trivial, i.e. $\mathcal{A} \ll \mathcal{A}'$ if and only if $\mathcal{A} = \cB$.\label{refx4}
\end{theo}

\begin{prsk}
One assumes $\mathcal{A} \neq \cB$ so that there exist distinct $x,y \in \mathbb{R}$ with $x \sim_\mathcal{A} y$. Then one picks a non-$\cB$-measurable set $T$ which separates $x$ and $y$. The finite subsets of $T$ and $\mathbb{R} \setminus T$ induce a directed set of finitely generated partitions whose supremum has as invariant sets only $T$ and $\mathbb{R} \setminus T$. Since these are non-$\cB$-measurable, the corresponding supremum of fgp-sub-$\sigma$-algebras is $\cB$. But all the partitions separate $x$ and $y$ hence the corresponding $\sigma$-algebras cannot be below $\mathcal{A}$.\qed
\end{prsk}

\begin{coro}
The way-below relation on $(\mathrm{Sm}(\mathbb{R}, \cB), \lessdot)$ is trivial.
\end{coro}

This is an unsatisfactory result, especially considering the fact that for any set $X$ the lattice $(\mathrm{Eq}(\mathbb{R}), \lessdot)$ of all equivalence relations of $X$ is algebraic, and thus from a domain-theoretical viewpoint carries a very well-behaved ordere structure.

\section{Conclusion}

For a given measurable space $(X, \mathcal{M})$ we have investigated the properties of the inclusion order and its opposite on the sets $\mathrm{Sub}_\sigma(\mathcal{M})$ (of sub-$\sigma$-algebras of $\mathcal{M}$), $\mathrm{Sub}^{\mathrm{eq}}_\sigma (\mathcal{M})$ (of equivalence-induced sub-$\sigma$-algebras of $\mathcal{M}$) and $\mathrm{Sub}^{\omega}_\sigma (\mathcal{M})$ (of countably-generated sub-$\sigma$-algebras of $\mathcal{M}$) from a domain-theoretical viewpoint. In particular we have shown that in the important case of the standard Borel-structure on the real numbers the way-below relation on these posets is trivial. Along the way we have also proved that the category of analytic spaces does not have all coequalizers, generalising Doberkat's result in \cite{eedconglatt} that it is not closed under pushouts in the category of measurable spaces.

For the inclusion order our results also hold for the $\w$-chain way-below relation. We do not know if this is the case for the opposite order investigated in section 4, as Theorem \ref{refx4} uses a directed set that cannot be reduced to an $\w$-chain. In fact, we also do not know whether for an analytic space an analogous result of Corollary \ref{refx5} holds for $\w$-chains. Such a result would be interesting as the nature of $\sigma$-algebras and smooth relations is inherently related to countable structures and thus $\w$-accessibility might provide a more successful approach than directed accessibility in this case.

\subsection*{Acknowledgments}

The author would like to thank Ernst-Erich Doberkat for his encouragement in doing this research. Helpful discussions with Christoph Schubert were appreciated.

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