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% Author Package file for use with AMS-LaTeX 1.2
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%\controldates{4-OCT-1996,4-OCT-1996,4-OCT-1996,11-OCT-1996}
\documentclass{era-l}
%\issueinfo{2}{2}{OCT}{1996}
\pagespan{98}{100}
\PII{S 1079-6762(96)00013-3}
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%\documentclass[12pt,a4paper]{amsart}

\newtheorem{thm}{Theorem}
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\newtheorem{cor}[thm]{Corollary}
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\begin{document}

\title{On the cut point conjecture}

\author{G. A. Swarup}

\address{ The University of Melbourne, Parkville, 3052, Victoria, Australia }


%\author{G. A. Swarup\\ 
%{\ }\lower .4cm\hbox{\centerline
%{\bf Dedicated to John Stallings}}\\
%{\ }\lower .2cm\hbox{\centerline{\bf on his 60$^{\text{th}}$ Birthday}}
%}
\dedicatory{Dedicated to John Stallings on his $60$th birthday}

\email{gadde@maths.mu.oz.au}

% for Latex2e and amsart use the following instead:
%\title[Cut point conjecture]{On the cut point conjecture}
%\author[Swarup]{ G. A. Swarup}
\date{June 4, 1996}

%\maketitle


\commby{Walter Neumann}

\subjclass{Primary 20F32; Secondary 20J05, 57M40}

\keywords{Gromov hyperbolic group, Gromov boundary, cut point, local
connectedness, dendrite, R-tree}

%\begin{document}

\begin{abstract}
We sketch a proof of the fact that the Gromov boundary of a hyperbolic group
does not have a global cut point if it is connected. This implies, by a
theorem of
Bestvina and Mess, that the boundary is locally connected if it is connected.
\end{abstract}


\maketitle

%\markboth{G.A. SWARUP}{ON THE CUT POINT CONJECTURE}

The object of this note is to sketch a proof of the following theorem:
\begin{thm}
If $\Gamma$ is a one ended hyperbolic group, then the Gromov boundary
$\partial \Gamma$ of $\Gamma$ does not have a global cut point.
\end{thm}

It seems that several people independently thought of using treelike
structures in this context, among them
Bill Grosso in an unpublished manuscript \cite{8}. 
The most significant advances
in this
direction were carried out by Brian Bowditch in a brilliant series of
papers (\cite{4}-\cite{7}). We draw
heavily from his work. He proved Theorem 1 in the case when $\Gamma$ is one
ended and does not split over a two ended group
\cite{6}, and in the case when $\Gamma$ is  strongly accessible and one ended
\cite{5}. Our strategy is to
relativize some of the arguments of \cite{4}, \cite{5}, 
use Levitt's construction \cite{9}
to obtain an
R-tree on which a subgroup of $\Gamma$ acts isometrically, and then use a
relative version of the
main theorem of \cite{2} to arrive at a contradiction. One of the
main results of \cite{4} asserts:
\begin{thm} [Bowditch \cite{4}]
If $\partial \Gamma $ has a global cut point, then $\partial \Gamma$ has a
nontrivial, equivariant
dendrite quotient $D(\partial \Gamma )$ on which $\Gamma$ acts as a
discrete convergence group.

\end{thm}

We refer to \cite{4} for definitions; a dendrite is a compact separable R-tree.
We assume that $\partial \Gamma$ has a global cut point, and we start with a
graph of group decomposition
of $\Gamma$ over two ended subgroups in which none of the vertex groups
splits over a finite or
two ended subgroup relative to the edge groups in it. We may assume that
the action of $\Gamma$ on the associated tree $\Sigma$ is minimal,
reduced, without inversions, and $\Sigma / {\Gamma}$ is compact. Since we
are assuming that $\partial \Gamma$
has a global cut point, such splittings of $\Gamma$ exist by \cite{6} 
and \cite{1}.


By \cite{5}, \cite{7}, there is a natural decomposition $\partial \Gamma
=\partial_{0}\Gamma \cup \partial_{\infty} \Gamma$, where
\[ %$$
\partial_{0}\Gamma = \bigcup_{v\in V(\Sigma)} \Lambda \Gamma(v)\] %$$ 
and $
\partial_{\infty} \Gamma$ is naturally identified with $\partial \Sigma$
($\Lambda G $ for a subgroup of $\Gamma$ denotes the limit set of $G$ in
$\partial \Gamma$ and
may be identified with $\partial G $ if $G$ is quasiconvex in $\Gamma$).
Our starting point was the observation that the end points of an edge group
cannot be separated by a global cut point, and
it follows from the description in \cite{4} of the quotient map $\partial
\Gamma\rightarrow  D(\partial \Gamma )$ that:
\begin{prop}
 The fixed points of $\Gamma (e)$, for each edge $e$, are identified in the
dendrite quotient $D(\partial \Gamma)$.
\end{prop}

This was first observed by the methods of \cite{10} and the generalized
accessibility of \cite{1}, but Bowditch pointed out that it
follows immediately from  Proposition 4 below. We now consider the images
in $D(\partial \Gamma)$ of $\Lambda(\Gamma (v)$, for $v\in V(\Sigma )$.
Bowditch considers for each directed edge $\vec{e}$ of $\Sigma$ a subset
$\Psi (\vec{e})$
of $\partial \Gamma$ defined  as follows. Let $\Phi (\vec{e})$ denote the
connected component of $\Sigma$ minus the interior of $e$
which contains the tail of $\vec{e}$. Then, $\Psi (\vec{e})$ is the part of
the limit set of $\Gamma$
to the left of $\vec{e}$:
\[ % $$
\Psi (\vec{e}) =\partial \Phi (\vec{e}) \cup \bigcup_{v \in V (\Phi
(\vec{e}))} \Lambda \Gamma(v).\] %$$
Let $\vec{\Delta} (v)$ denote the set of directed edges of $\Sigma$ with
head at $v$.

\begin{prop} [Bowditch, Lemma 3.3 and Proposition 5.1 of \cite{5}] With the
notation above, we have
\[ % $$
\Psi (\vec{e}) \cup \Psi (-\vec{e})=\partial \Gamma =\Lambda \Gamma
(v)\cup \bigcup_{\vec{e}\in \vec{\Delta} (v)}\Psi(\vec{e}),\] %$$
\[ %$$
\Psi (\vec{e}) \cap \Psi (-\vec{e})=\partial \Gamma (e),\] %$$
and $\Psi(\vec{e})$ is closed, $\Gamma (e)$ invariant, and connected.
\end{prop}

The fact that $\Psi (\vec{e})$ is connected requires some argument. Since
$\Psi (\vec{e})$ and $\partial \Gamma$ are connected, it follows that by
identifying the two points in
$\Lambda \Gamma (e)$ for $e \ni v$ we obtain a connected quotient of
$\partial \Gamma (v)$.
This together with Proposition 3 shows that the image of $\Lambda \Gamma
(v)$ in $D(\partial \Gamma)$
is connected for each vertex $v$ of $\Sigma$. Denote this by $D(M(v))$.
At least one of
these must be nontrivial (not a point) from the 
decomposition of $\partial \Gamma$
described before Proposition 3.
Choose one such vertex $v$; we denote the quotient $D(M(v))$ by $D(M)$,
$\Gamma (v)$ by $G$,
and call $\Gamma(e)$ for $e\ni v$ the peripheral subgroups of $G$.
Restricting the action of $\Gamma$ to $G$
we see that:

\begin{prop}$ G $ acts as a discrete convergence group on $D(M)$, and the
peripheral subgroups of $G$ fix points of $M$.
Moreover, $D(M)$ is a nontrivial dendrite.
\end{prop}

 By removing the terminal points of
 $D(M)$ one obtains as in \cite{4} an R-tree $T$ on which the induced action of
$G$ is nonnesting
 (see Section 7 of \cite{4}; nonnesting means that for any compact interval
$A$, if $g(A) \subset A$,
then $g(A)=A$). Since the action of $G$ on $D(M)$ is a discrete
convergence group action,
the edge stabilizers for the action on $T$ are finite. A peripheral
subgroup $H$ of $G$ fixes a
  point of $D(M)$ and thus fixes either a point of $T$ or exactly one end
of $T$.
Since a nonnesting action on a real tree does not have parabolic elements
(in the
usual sense; see \cite{9}), it follows that the peripheral subgroups of $G$ fix
points of $T$.
In summary, we have:
\begin{prop} If $\partial \Gamma$ has a global cut point, then some vertex
group
$\Gamma (v)=G$ admits a nonnesting action on an R-tree $T$ with finite edge
stabilizers and such that each peripheral subgroup of $G$ fixes some point
of $T$.
\end{prop}

Under these conditions Levitt (\cite{9}, see also \cite{6} --- perhaps it is
possible to use \cite{6} too for the rest of the argument) constructs an action
of $G$ by
isometries
on an R-tree $T_0$ such that a subgroup of $G$ fixing an arc of $T_0$ also
fixes an arc of $T$.
The construction is natural, and since we have only finitely many conjugacy
classes of peripheral subgroups,
we see that the peripheral subgroups of $G$ still fix points of $T_0$ (this
needs enlarging $K$
in Levitt's construction; see Corollary 6 of \cite{9}). Thus we have the
analogue of Proposition 6 with the action of $G$ by isometries rather than
just homeomorphisms.
The condition on edge stabilizers shows that the action is stable in the
sense of
Bestvina and Feighn \cite{2}. Thus by the relative version of the main theorem
of \cite{2}
(this is Theorem 9.6 of \cite{2}), we conclude that there is a  nontrivial
decomposition of $G$ along a virtually cyclic group such that each
peripheral subgroup
is conjugate to a subgroup of a vertex group. Since we started with a
$G=\Gamma (v)$
which does not admit such a splitting, we have the desired contradiction to
complete the
proof of Theorem 1. By \cite{3}, it follows that $\partial \Gamma$ is locally
connected and the theory of \cite{7}
goes through for all hyperbolic groups. This leads to a precise
characterization of when $\operatorname{Out}(\Gamma)$ is infinite.

%{\bf Acknowledgement:} 
\section*{Acknowledgement} I am grateful to Mladen Bestvina, Brian Bowditch,
Mark Feighn, Gilbert Levitt, and Chuck Miller
for their help in the preparation of this note. Of course, it would not
have been possible to write this note
without Bowditch's outstanding work.

\begin{thebibliography}{99}

\bibitem{1}  M. Bestvina and M. Feighn, \emph{Bounding the complexity 
of simplicial actions on trees},
Inv. Math. \textbf{103} (1993), 449--469.
 \MR{92c:20044}
\bibitem{2}  
\bysame, \emph{Stable actions of groups on real
trees}, Inv. Math. \textbf{121} (1995), 287-361.
 \MR{96h:20056}
\bibitem{3}  M. Bestvina and G. Mess, \emph{The boundary of negatively curved
groups}, J. Amer. Math. Soc. \textbf{4} (1991), 469--481.
\MR{93j:20076}
\bibitem{4}  B. H. Bowditch, \emph{Treelike structures 
arising from continua and
convergence groups}, Preprint (1995).
\bibitem{5}  
\bysame, \emph{Boundaries of strongly accessible groups},
Preprint (1996).
\bibitem{6}  
\bysame, \emph{Group actions on trees and dendrons}, Preprint (1995).
\bibitem{7}  
\bysame, \emph{Cut points and canonical splittings of hyperbolic
groups}, Preprint (1995).
\bibitem{8}  W. Grosso, Unpublished  manuscript, Berkeley (1995).
\bibitem{9}  G. Levitt, \emph{Nonnesting actions on real trees}, 
Preprint (1996).
\bibitem{10}  G. P. Scott and G. A. Swarup, \emph{An algebraic 
annulus theorem}, Preprint (1995).

\end{thebibliography}


\end{document}