• Boundaries
• Section2

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

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do

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-dimensional stable Menger spaces appear as boundaries?

Example 2.1 [Damian Osajda]\label{bldgs} Let

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be a thick right-angled hyperbolic building of rank

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, i.e. with apartments isometric to

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. Then the ideal boundary of

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is a stable Menger space

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. However

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-dimensional right-angled hyperbolic reflection groups exist only for

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.

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

Background: The Menger space

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is obtained by iteratively subdividing an

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-cube into

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subcubes and removing those that do not touch the

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-skeleton, see MR920964 for a detailed discussion of the topology of these spaces. Below are few properties of

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:

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   has topological dimension

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   .

2. dvipng error! exitcode was 2 (signal 0), transscript follows:
   
   

   is stable when

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   (that is, replacing

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   by a larger value does not change

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   ).

3. Any

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   -dimensional compact metric space embeds in some stable

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   .

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

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appear as boundaries of hyperbolic Coxeter groups, see MR1684267. According to work of Misha Kapovich and Bruce Kleiner MR1834498: if

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is 1-dimensional, connected and has no local cut points, then

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is homeomorphic to a Sierpinski carpet (

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) or the Menger space

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.

Problem 2.4 (Comments) [Mike Davis] Are there torsion-free hyperbolic groups

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with

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?

Background: Here

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is the cohomological dimension over a ring

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. Mladen Bestvina and Geoff Mess MR1096169 have shown that:

a. For torsion-free hyperbolic groups

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.

b. There are hyperbolic groups

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such that

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and

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.

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

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which is hyperbolic relative to some parabolic subgroups that are nilpotent of class

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whose Bowditch boundary is homeomorphic to some

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-sphere?

Remark [Tadeusz Januszkiewicz] Strict hyperbolization of piecewise linear manifolds gives many examples of hyperbolic groups

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with

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homeomorphic to

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.

Problem 2.7 (Comments) [Misha Kapovich] Suppose that

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is a compact metrizable topological space,

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is a convergence action which is topologically transitive, i.e. each

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--orbit is dense in

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. Is there a Gromov-hyperbolic space

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with the ideal boundary

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so that the action

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extends to a uniformly quasi-isometric quasi-action

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?

Background: Suppose that

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is a topological space,

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is the set of triples of distinct points in

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. The space

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has a natural topology induced from

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. A topological group action

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is called a convergence action if the induced action

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is properly discontinuous. A convergence action

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is called uniform if

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is compact. Examples of convergence group actions are given by uniformly quasi-Moebius actions

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, e.g. are induced on

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by uniformly quasi-isometric quasi-actions

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. Brian Bowditch MR1602069 proved that each uniform convergence action

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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

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cubical complexes.

Background. A

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cubical complex is a

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complex

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where every

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-cell is a combinatorial cube, isometric to a polytope in

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, 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

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cannot be homeomorphic to

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. Moreover,

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cannot contain an essential

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-sphere for

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.

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