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3. 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.
    1. Problem 3.1.

      [Panos Papasoglu] 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?
        1. 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 [MR1638764] and Swarup [MR1412948] showed that the boundary of a one ended hyperbolic group has no cut points. Bowditch [MR1638764] showed that if the boundary has a local cut point then the group splits over a $2$-ended group.
            • Problem 3.2.

              [Panos Papasoglu] 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 $G$?
                1. 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 ([MR2153400], [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".
                    • Problem 3.3.

                      [Geoghegan] Are one-ended CAT(0) groups semi-stable at infinity?
                        1. Remark. The no cut point theorem for hyperbolic groups implies (via work of Bestvina-Mess [MR1096169]) that hyperbolic groups are semi-stable at infinity. Although it is known now [arXiv:math.GR/0701618] that CAT(0) boundaries have no cut points the semi-stability question is still open for CAT(0) groups.
                            • Problem 3.4.

                              Are splittings of one-ended finitely generated groups over 2-ended groups invariant under quasi-isometries?
                                1. Remark. The answer is positive for finitely presented groups [MR2153400]
                                    • Problem 3.5.

                                      [Kleiner] 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?
                                        1. Remark. It follows by work of Bowditch [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.
                                            • 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-$\mathbb Z ^2$?
                                                1. Remark. One might ask more generally whether JSJ decompositions over virtually-$\mathbb Z ^2$ groups are invariant under quasi-isometries.
                                                    • Problem 3.7.

                                                      [Panos Papasoglu] 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-$\mathbb Z ^{n-2}$ ?
                                                        1. 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 [MR1998479] and [MR2300450].
                                                            •     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:

                                                              Problem 3.8.

                                                              [Panos Papasoglu] Let $X$ be a locally connected homogeneous continuum of dimension $\geq 2$. Is it true that $X$ is not separated by an arc?
                                                                1. Remark. The answer is positive in the case of simply connected continua ([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 ([MR0096181]) while boundaries of hyperbolic groups of dimension 1 were classified by Kapovich-Kleiner ([MR1834498]). One does not know much in either case when dimension is greater than 1.

                                                                      Cite this as: AimPL: JSJ Decompositions, available at http://aimpl.org/jsjdecomp.