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\begin{document}
\title{JSJ Decompositions}
\author{Edited by Panos Papasoglu}

\urlstub{jsjdecomp}

\maketitle

Splitting theory of groups parallels decomposition theorems in
3-manifold theory. Kneser's theorem has its analog in Grushko's
theorem, Haken finiteness and hierarchies gave rise to
accessibility and strong accessibility in group theory while JSJ
decompositions of 3-manifolds were shown recently to extend to
group theory too. One should stress that these "generalizations"
from 3-manifolds to groups were by no means routine and a lot of
questions remain open today. The structure theory that one obtains
for groups is often much more general that the 3-manifold theory
and it had many striking applications (isomorphism problem for
hyperbolic groups, Tarski problems etc). We refer to
\cite{MR1469317} and \cite{MR1664694} for the basic definitions.



\section{JSJ Decompositions}





\begin{problemblock}

\begin{problem}[1.1] Is there a JSJ decomposition for finitely presented groups over small groups? \end{problem}




\begin{remark}Dunwoody \cite{Du}
has announced recently a much more general result for groups
acting on $\mathbb R$-trees that seems to imply the existence of JSJ
decompositions over small groups. Guirardel and Levitt have also
proposed a generalized JSJ theory in the spirit of outer space
\cite{MR2319455}, \cite{GL}.\end{remark}

\begin{remark}The first case to look at is the case of
decompositions over solvable Baumslag-Solitar groups. If such a
decomposition exists then the edge groups of the decomposition
won't necessarily be small (see \cite{MR2221253}). On the other
hand there is a natural conjecture for the enclosing groups: one
expects them to be fundamental groups of complexes of groups where
the underlying complex is a square complex homeomorphic to a
surface and all edges and faces are stabilized by a fixed group
$F$ (the fiber). However the homomorphisms from edge groups to
face groups are not necessarily isomorphisms. \end{remark}

\end{problemblock}



\begin{problemblock}We call a splitting of $G$ over $C$ \emph{unfolded} if $G$
does not split over any proper subgroup of $C$.

\begin{problem}[1.2] Let $G$ be a finitely presented group that does not  
split over a virtually abelian group. Can $G$ have infinitely many unfolded splittings over distinct subgroups isomorphic to ${\mathbb F}_2$? \end{problem}




\begin{remark}A splitting of a group $G$ over a group $C_1$ is called elliptic
with respect to a splitting of $G$ over $C_2$ if $C_1$ fixes a
point of the Bass-Serre tree of the splitting over $C_2$.
Otherwise it is called hyperbolic.

A pair of splittings of $G$ over $C_1,C_2$ can be
elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic,
hyperbolic-hyperbolic.

A crucial observation for the JSJ theory of splittings of 1-ended
finitely presented groups over $\mathbb Z$ is that any two splittings
over infinite cyclic groups $C_1,C_2$ are either elliptic-elliptic
or hyperbolic-hyperbolic.

The Bestvina-Feighn accessibility theorem (\cite{MR1091614}) gives
a bound on the number of elliptic-elliptic splittings over $\mathbb Z$ 
(and more generally over small groups). So the main issue for
the JSJ theory is understanding hyperbolic-hyperbolic splittings.
In the case of $\mathbb Z$-splittings it is easy to produce
infinitely many distinct hyperbolic-hyperbolic splittings, just
consider the splittings corresponding to simple closed curves on a
surface. The JSJ theory says that in fact this is the only way in
which such splittings arise. So the problem above asks whether
there are families of hyperbolic-hyperbolic splittings over the
free group of rank 2, ${\mathbb F}_2$.
\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.3]  Is the isomorphism problem solvable for the graphs of
groups for which all edges and vertices are labelled by $\mathbb Z$? \end{problem}




\begin{remark}One of the main applications of the JSJ theory was
the solution of the isomorphism problem for hyperbolic groups. In
general however there is no canonical JSJ decomposition (see
though \cite{MR2032389}) and one does not have a good description
of the set of all JSJ decompositions. The simplest case for which
this question is open is that of graphs of groups where all vertex
and edge groups are isomorphic to $\mathbb Z$. In this case an
algorithmic description is equivalent to the isomorphism problem
stated above. Levitt remarks that the isomorphism problem for
graphs of groups where all edges and vertices are labelled by
$\mathbb Z ^4$ is unsolvable.\end{remark}

\end{problemblock}



\begin{problemblock}A Dehn twist of $G$ is an automorphism $\phi $ of the
following form: either $G$ splits as $A*_CB$, $t\in C$ is central,
$\phi (a)=a$ for $a\in A$ and $\phi (b)=tbt^{-1}$ for $b\in B$; or
$G$ is an HNN extension $A*_C$, with stable letter $s$, $t\in C$
is central, $\phi (a)=a$ for $a\in A$ and $\phi (s)=ts$.

\begin{problem}[1.4]  Let $G$ be a $CAT(0)$ group with $Out(G)$
infinite. Is it true that $G$ admits a Dehn twist of infinite
order? \end{problem}




\begin{remark}Does G even have any nontrivial splittings?\end{remark}

\end{problemblock}



\section{Accessibility Questions}

We recall the definition of strong accessibility of a group $G$
over a family of subgroups $\mathcal C$: decompose $G$ as a graph of
groups with edge groups in $\mathcal C$. Then decompose the vertex groups
of this graph as graphs of groups with edge groups in $\mathcal C$ and so
on. If there is such a series of decompositions that terminates
(i.e. at some stage the vertex groups do not admit any
decomposition over $\mathcal C$) then the group is called
\emph{strongly accessible} over $\mathcal C$. We note that one asks
only for the existence of such a series of decompositions and not
that \emph{every} such series terminates. This is analogous to
hierarchies for 3-manifolds.



\begin{problemblock}

\begin{problem}[2.1] Let ${\mathcal C}$ be the class of finite and 
virtually-$\mathbb Z$ groups. Are hyperbolic groups strongly accessible over ${\mathcal C}$?  \end{problem}




\begin{remark}Grushko's theorem implies that finitely generated
groups are strongly accessible over the trivial group. Dunwoody's
theorem \cite{MR807066} implies that finitely presented groups
are strongly accessible over ${\mathcal C}=\{\text {finite groups}\}$.
Delzant and Potyagailo \cite{MR1838998} showed that hyperbolic
groups with no 2 torsion are strongly accessible over the class
${\mathcal C}$  of finite and 2-ended groups. So the question remaining
here is whether the no-2 torsion assumption can be removed.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[2.2] Let ${\mathcal C}$ be the class of finite and virtually-$\mathbb Z$
groups. Are finitely presented groups strongly accessible over
${\mathcal C}$? \end{problem}




\begin{remark}One can ask this more generally for splittings over
small groups.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[2.3] Is there a 1-ended finitely presented group $G$ such
that $G\simeq G*_{\mathbb Z}$? \end{problem}




\end{problemblock}



\begin{problemblock}One may ask more generally

\begin{problem}[2.4]  Let ${\mathcal C}$ be the class of finitely generated groups.
Are finitely presented groups strongly accessible over ${\mathcal C}$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.5] Let $G=A*_{C_n}B_n,\, n\in \mathbb N$ where $G$ is a
finitely generated, $C_n$ are slender and $C_n < C_{n+1}$ for all
$n$. Is it true that $G=A*_C B$ where $C < C_n$ for all $n$? Similarly 
for HNN extensions.  \end{problem}




\begin{remark}This was shown by Rips-Sela \cite{MR1469317} for
$C_n$ isomorphic to $\mathbb Z$. Weidmann pointed out that one may
pose the question even more generally omitting `finitely
generated' for $G$. It is true for finitely presented groups
\cite{MR2221253}.\end{remark}

\end{problemblock}



\section{Group splittings and asymptotic topology}

One may pose a very general question: What meaningful geometric
properties distinguish Cayley graphs from arbitrary metric spaces?
Here geometric of course is in the sense of Gromov's
quasi-isometries. A lot of the recent work in geometric group
theory falls in this frame. Stallings' ends theorem, Gromov's
polynomial growth theorem and Varopoulos isoperimetric inequality
are three deep positive results distinguishing Cayley graphs from
general metric spaces. On the other hand one may see Grigorchuk's
intermediate growth examples and
Rips-Sapir-Birget-Olshanskii-Bridson isoperimetric inequalities
examples as negative results to the above question, saying that
"everything is possible" for Cayley graphs.

Splitting theory has provided some further positive results in
this quest and there is some reasonable hope for further progress.
We state here some specific questions in this spirit.




\begin{problemblock}

\begin{problem}[3.1]  Let $G$ be a hyperbolic group such that
$\partial G$ is not separated by a Cantor set. Is it true that $\partial G$ is
not separated by a simple path? 


 \end{problem}




\begin{remark}Since Stallings ends theorem the theory of splittings
has been related to "asymptotic topology" of Cayley graphs. The
relation became clearer after Bowditch's work. In the case of
hyperbolic groups the asymptotic topology of the group is
reflected on the topology of the boundary which is a compact
metrizable space. Bowditch \cite{MR1638764} and Swarup
\cite{MR1412948} showed that the boundary of a one ended
hyperbolic group has no cut points. Bowditch \cite{MR1638764}
showed that if the boundary has a local cut point then the group
splits over a $2$-ended group.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.2] Let $G$ be a finitely generated group which does not split over a finite group.
Is it true that (any subset of) a quasi-ray does not separate
coarsely the Cayley graph of $G$? \end{problem}




\begin{remark}By quasi-ray we mean a uniform embedding of $\mathbb R^+$ 
in the Cayley graph of $G$. We remark that the answer to the
above question is positive for finitely presented groups (\cite{MR2153400}, \cite{Kl}).
This question is similar to Bowditch's no
cut point theorem for finitely generated groups. We recall below
the relevant definitions.

A map $f:X\to Y$ between two metric spaces is called a uniform
embedding if

 i) there are $C,D$ such that $d(f(x),f(y))\leq Cd(x,y)+D$ for all $x,y\in X$

 ii) if $d(x_n,y_n)\to \infty $ then $d(f(x_n),f(y_n))\to \infty $ for any two sequences $(x_n),(y_n)$ in $X$.


We say that a set $A$ is coarsely contained in a set $B$ if $A$ is
contained in a finite neighborhood, $N_K(B)$, of $B$.

We say that a subset $Y$ of $X$ coarsely separates $X$ if for some
finite neighborhood $N_K(Y)$, $X-N_K(Y)=X_1\sqcup X_2$ with
$X_1,X_2$ open and neither $X_1$ nor $X_2$ is coarsely contained
in $Y$.

It is worth remarking that the no quasi-ray separates question can
be seen also as generalizing Stallings theorem. Indeed one may
restate Stallings theorem as follows:

Let $G$ be a finitely generated group
that does not split over a finite group. Then a point does not
separate coarsely the Cayley graph of $G$. 

So if the answer to the
above problem is positive it gives a strengthening of Stallings
ends theorem. We note also that the exact analog of the no-cut
point theorem would be to show that quasi-"horo-balls" do not
separate, rather than quasi-rays. This is not known for finitely
presented groups though, in fact it is not even clear what is the
right notion of quasi-"horo-ball".
\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.3] Are one-ended CAT(0) groups semi-stable at infinity? \end{problem}




\begin{remark}The no cut point theorem for hyperbolic groups
implies (via work of Bestvina-Mess \cite{MR1096169}) that
hyperbolic groups are semi-stable at infinity. Although it is
known now \cite{arXiv:math.GR/0701618} that CAT(0) boundaries have no
cut points the semi-stability question is still open for CAT(0)
groups.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.4] Are splittings of one-ended finitely generated groups
over 2-ended groups invariant under quasi-isometries? \end{problem}




\begin{remark}The answer is positive for finitely presented groups
\cite{MR2153400}\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.5] Let $G$ be a finitely generated group such
that there is a sequence of quasi-circles that separate its Cayley
graph. Is $G$ a virtually surface group?  \end{problem}




\begin{remark}It follows by work of Bowditch \cite{MR2099199} that
this is true for finitely presented groups. By a sequence of
quasi-circles that separate we mean the following: we take a union
of circles $X=\bigcup \{C_n \}$ in $\mathbb R^2$ with radii tending
to infinity and we consider a uniform embedding $f$ from $X$ to
the Cayley graph of $G$. Now we assume that for some fixed $N$ the
$N$-neighborhood of $f(C_n)$ separates the Cayley graph and at
least 2 components are not contained in the $n$ neighborhood of
$f(C_n)$.

For example in the hyperbolic or Euclidean plane one can take
$C_n$ to be a sequence of circles (i.e. boundaries of balls). In
the case of the Euclidean plane these are quasi-isometrically
embedded but this is not the case for the hyperbolic plane.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.6]  Are splittings over virtually-$\mathbb Z ^2$ groups
invariant under quasi-isometries? More precisely let $G$ be a
one-ended finitely presented group that does not split over a
2-ended group. Suppose that $G$ splits over virtually-$\mathbb Z ^2$.
Is it true a group quasi-isometric to $G$ splits over
virtually-$\mathbb Z ^2$? \end{problem}




\begin{remark}One might ask more generally whether JSJ
decompositions over virtually-$\mathbb Z ^2$ groups are invariant
under quasi-isometries.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[3.7] Let $G$ be a group with $asdim\,(G)\geq n$, $n>3$. 
Assume that a uniformly embedded copy of $\mathbb Z ^{n-2}$ 
separates coarsely the Cayley graph of $G$. Is it true
then that $G$ splits over virtually-$\mathbb Z ^{n-2}$ ? \end{problem}




\begin{remark}Both in Stallings theorem and in Bowditch JSJ theory
there are some classes of groups that can be thought of as
"exceptional". In Stallings theorem these are the groups with 2
ends (all commensurable to $\mathbb Z$) and in Bowditch's theorem it
is hyperbolic triangle groups (although their boundary has local
cut points they do not split). As one tries to generalize these
theorems to splittings over $\mathbb Z ^n$ it is natural to expect
that the number of "exceptions" increases. The question above
avoids this technical issue by formulating the problem so that
exceptions are ruled out.

Some evidence in favor of this is provided by \cite{MR1998479} and
\cite{MR2300450}. \end{remark}

\end{problemblock}



\begin{problemblock}It is interesting to note that some of the above questions can be
stated also for locally connected continua (especially homogeneous
continua). For example the question on quasi-rays that separate
has the following twin:

\begin{problem}[3.8] Let $X$ be a locally connected
homogeneous continuum of dimension $\geq 2$. Is it true that $X$ is
not separated by an arc?  \end{problem}




\begin{remark}The answer is positive in the case of simply
connected continua (\cite{math.GN/0611817}) which heuristically
corresponds to the case of finitely presented groups. Several
people have noted affinities between continua and groups. It is
worth noting that locally connected homogeneous continua of
dimension 1 were classified by Anderson (\cite{MR0096181}) while
boundaries of hyperbolic groups of dimension 1 were classified by
Kapovich-Kleiner (\cite{MR1834498}). One does not know much in
either case when dimension is greater than 1.\end{remark}

\end{problemblock}





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\end{document}
