Boundaries/Section2 - Problems in Geometric Group Theory

2 Topology of boundaries of hyperbolic groups

Problem 2.1 (Comments) [Misha Kapovich] What spaces can arise as boundaries of hyperbolic groups? As a sub-problem: For which $k$ do $k$ -dimensional stable Menger spaces appear as boundaries?

Example 2.1 [Damian Osajda]\label{bldgs} Let $X$ be a thick right-angled hyperbolic building of rank $n+1$ , i.e. with apartments isometric to $\H^{n+1}$ . Then the ideal boundary of $X$ is a stable Menger space $M_{n,k}$ . However $n+1$ -dimensional right-angled hyperbolic reflection groups exist only for $n\le 3$ .

Problem 2.2 (Comments) Can one remove the "right-angled" assumption in Osajda result?

Background: The Menger space $M_{k,n}$ is obtained by iteratively subdividing an $n$ -cube into $3^n$ subcubes and removing those that do not touch the $k$ -skeleton, see MR920964 for a detailed discussion of the topology of these spaces. Below are few properties of $M_{k,n}$ :

$M_{k,n}$ has topological dimension .
$M_{k,n}$ is stable when $n\ge 2k+1$ (that is, replacing by a larger value does not change $M_{k,n}$ ).
Any -dimensional compact metric space embeds in some stable $M_{k,n}$ .

Problem 2.3 (Comments) [Panos Papasoglu] What 2-dimensional spaces arise as boundaries of hyperbolic groups? Can restrict to cases with no virtual splitting, no local cut points or cut arcs, and no Cantor set that separates.

Background: 2-dimensional Pontryagin surfaces and 2-dimen\-sional Men\-ger spa\-ces $M_{2,5}$ appear as boundaries of hyperbolic Coxeter groups, see MR1684267. According to work of Misha Kapovich and Bruce Kleiner MR1834498: if $\geo G$ is 1-dimensional, connected and has no local cut points, then $\geo G$ is homeomorphic to a Sierpinski carpet ( $M_{1,2}$ ) or the Menger space $M_{1,3}$ .

Problem 2.4 (Comments) [Mike Davis] Are there torsion-free hyperbolic groups $G$ with $cd_\Q(G) / cd_\Z(G) < 2/3\quad$ ?

Background: Here $cd_R$ is the cohomological dimension over a ring $R$ . Mladen Bestvina and Geoff Mess MR1096169 have shown that:

a. For torsion-free hyperbolic groups $cd_R(G)=cd_R(\geo G)+1$ .

b. There are hyperbolic groups $G$ such that $cd_\Z(G)=3$ and $cd_\Q(G)=2$ .

Problem 2.5 (Comments) [Nadia Benakli] What can be said about boundaries arising from strict hyperbolization constructions of Charney and Davis, MR1318879?

Problem 2.6 (Comments) [Ilia Kapovich] Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\ge 3$ whose Bowditch boundary is homeomorphic to some $n$ -sphere?

Remark [Tadeusz Januszkiewicz] Strict hyperbolization of piecewise linear manifolds gives many examples of hyperbolic groups $G$ with $\partial_\infty G$ homeomorphic to $S^n$ .

Problem 2.7 (Comments) [Misha Kapovich] Suppose that $Z$ is a compact metrizable topological space, $G\acts Z$ is a convergence action which is topologically transitive, i.e. each $G$ --orbit is dense in $Z$ . Is there a Gromov-hyperbolic space $X$ with the ideal boundary $Z$ so that the action $G\acts Z$ extends to a uniformly quasi-isometric quasi-action $G\acts X$ ?

Background: Suppose that $Z$ is a topological space, $Z^{(3)}$ is the set of triples of distinct points in $Z$ . The space $Z^{(3)}$ has a natural topology induced from $Z^3$ . A topological group action $G\acts Z$ is called a convergence action if the induced action $G\acts Z^{(3)}$ is properly discontinuous. A convergence action $G\acts Z$ is called uniform if $Z^{(3)}/G$ is compact. Examples of convergence group actions are given by uniformly quasi-Moebius actions $G\acts Z$ , e.g. are induced on $Z=\geo X$ by uniformly quasi-isometric quasi-actions $G\acts X$ . Brian Bowditch MR1602069 proved that each uniform convergence action $G\acts Z$ is equivalent to the action of a hyperbolic group on its ideal boundary.

Problem 2.8 (Comments) [Tadeusz Januszkiewicz] Find topological restrictions on the ideal boundaries of $CAT(-1)$ cubical complexes.

Background. A $CAT(-1)$ cubical complex is a complex $X$ where every $n$ -cell is a combinatorial cube, isometric to a polytope in $\H^n$ , so that the isometry preserves the combinatorial structure. For instance, such a complex can cover closed hyperbolic 3-manifold. It was proven by Januszkiewicz and {\'S}wi{a}tkowski JS that $\geo X$ cannot be homeomorphic to $S^4$ . Moreover, $\geo X$ cannot contain an essential $k$ -sphere for $k\ge 4$ .

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