Boundaries/Section9 - Problems in Geometric Group Theory

9 Problems related to Cannon's Conjecture

Problem 9.1 (Comments) [Cannon's Conjecture, Version I] If

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is a (Gromov) hyperbolic group with

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homeomorphic to

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, then

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acts geometrically on

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Problem 9.2 (Comments) [Cannon's Conjecture, Version II] Under the same assumptions on

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Remark Perelman's proof of Thurston's geometrization conjecture implies that the Cannon's conjecture is equivalent to the finding that such

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is commensurable to a 3-manifold group: There exists an exact sequence

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with

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finite and

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Remark If

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is hyperbolic and torsion-free then

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iff

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is a

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group (a 3-dimensional Poincar\'{e} duality group), see MR1096169.

Problem 9.3 (Comments) [Conjecture of C.T.C. Wall] Every

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group is a 3-manifold group.

Problem 9.4 (Comments) [Cannon's Conjecture, Relative Version I] If

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is hyperbolic and

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is homeomorphic to the Sierpinski carpet, then

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acts geometrically on a convex subset of

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Remark This follows from the Cannon's Conjecture by doubling.

Problem 9.5 (Comments) [Cannon's Conjecture, Relative Version II] For

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hyperbolic relative to

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for a collection of virtually-

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subgroups

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, if the Bowditch boundary

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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latex error! exitcode was 1 (signal 0), transscript follows:

This is TeX, Version 3.14159 (Web2C 7.4.5)
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LaTeX2e <2001/06/01>
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, then

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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is commensurable to the fundamental group of a hyperbolic 3-manifold of finite volume. \label{another}

Remark The same problem could be posed allowing the boundary to be

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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or Sierpinski carpet.

Problem 9.6 (Comments) [Cannon's Conjecture, Analytic Version] If

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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is a hyperbolic group with

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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homeomorphic to the Sierpinski carpet, then the visual metric on

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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is quasisymmetric to some round Sierpinski metric:

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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Recent work of Mario Bonk gives simplifications and partial answers here.

Problem 9.7 (Comments) Prove Cannon's conjecture under additional assumptions, such as

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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, in Haken case (without appealing to Thurston's proof of the hyperbolization theorem)

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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latex error! exitcode was 1 (signal 0), transscript follows:

This is TeX, Version 3.14159 (Web2C 7.4.5)
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LaTeX2e <2001/06/01>
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group that splits over a surface group

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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acts on a CAT(0) cube complex

....

Remark Cannon's Conjecture is known for Coxeter groups

latex error! exitcode was 1 (signal 0), transscript follows:

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(a work by Mario Bonk and Bruce Kleiner). This follows of course from Andreev's theorem, but the point here is to give a proof which only uses the geometry of the ideal boundary of

latex error! exitcode was 1 (signal 0), transscript follows:

This is TeX, Version 3.14159 (Web2C 7.4.5)
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Problem 9.8 (Comments) [Misha Kapovich] Give positive solution to Problem~\ref{another} assuming the absolute case by doing "hyperbolic Dehn surgery" (see Groves--Manning, Osin). Namely, add relators

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latex error! exitcode was 1 (signal 0), transscript follows:

This is TeX, Version 3.14159 (Web2C 7.4.5)
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, where

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. For sufficiently long elements

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the quotient

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are known to be hyperbolic.

a. Prove that if

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's are sufficiently long then

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This is TeX, Version 3.14159 (Web2C 7.4.5)
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(/usr/share/texmf/tex/latex/base/article.cls
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is an (absolute)

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group, by, say, computing

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b. Assuming that each

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is a 3-manifold group, show that

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is a 3-manifold group as well.

To motivate a possible approach to Cannon's conjecture recall the following:

Theorem 9.1 [M.~Bonk, B.~Kleiner, MR1949785] Suppose that

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is a hyperbolic group,

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is a uniformly quasi-Moebius action on a metric space which is topologically conjugate to the action of

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on its ideal boundary. Assume that

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is Ahlfors

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-regular and has topological dimension

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. Then

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is quasi-symmetric to the round

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-sphere. In particular,

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acts geometrically on

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Therefore, given a hyperbolic group

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with

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homeomorphic to

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one would like to replace the visual metric

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with a quasisymmetrically equivalent one, which has Hausdorff dimension

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. Since Hausdorff dimension (

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) of a metric compact homeomorphic to

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, one could try to minimize Hausdorff dimension in the quasi-conformal gauge of

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, i.e. the collection

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of metric spaces

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which are quasisymmetric to

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. This motivates the following:

Definition 9.2 For a metric space

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, define its Pansu conformal dimension

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Likewise, Ahlfors regular Pansu conformal dimension of

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The importance of the latter comes from

Theorem 9.3 [M.~Bonk, B.~Kleiner, MR2116315] Suppose that

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is a hyperbolic group,

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is homeomorphic to

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. If the

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is attained then

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acts geometrically on

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Remark Bourdon--Pajot examples MR1979183 show that the

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for the boundaries of hyperbolic groups is not always attained.

Generally,

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attained iff there is a Loewner metric in

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(which is then minimizing).

Problem 9.9 (Comments) [Conjecture of Bruce Kleiner] For a hyperbolic group

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where the infimum is taken over all geometric actions of

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on metric spaces

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. A bolder conjecture would be that, when the infimum is attained, it is attained by a visual metric.

Problem 9.10 (Comments) What is

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of the standard Sierpinski carpet? In particular, does the above conjecture hold?

Problem 9.11 (Comments) [Juha Heinonen] Under what assumptions on hyperbolic groups

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with

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-Loewner boundary

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does it admit a 1-Poincar\'{e} inequality for the boundary?

Cannon's conjecture has a generalization to nonuniform convergence group actions on compacts. Here is one of such

Suppose

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is the support of a measured lamination on a surface

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and

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consists of topological disks. Lift this lamination to a lamination

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in the unit disk

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and define the following equivalence relation

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1. The closure of each component of

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in the closed disk

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is an equivalence class.

2. If

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is a geodesic which is not on the boundary of a component of

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, then the closure of

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is an equivalence class. The rest of the points of

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are equivalent only to themselves.

Note that

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-invariant and that the equivalence classes are cells. Therefore the quotient

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is homeomorphic to

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and the group

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acts on

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by homeomorphisms. One can check that this is a convergence group action.

More generally, one can form an equivalence relation using a pair of transversal laminations and make the corresponding

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--invariant quotient.

Problem 9.12 (Comments) \label{ct} [Cannon--Thurston] Is this action conjugate to a conformal action?

The situation here is, in many ways, more complicated than in Cannon's conjecture since there is no a priori a useful metric structure on

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. It is not even clear that there exists 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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One can reformulate this problem using theory of Kleinian groups as follows. According to the Ending Lamination Conjecture, there exists a discrete embedding

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so that the ending lamination of

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Problem 9.13 (Comments) \label{ct1} The limit set of the Kleinian group

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is locally connected.

In the presence of two geodesic laminations, the limit set of

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is the entire 2-sphere, so local connectedness is meaningless. Then the appropriate reformulation is as follows:

Problem 9.14 (Comments) \label{ct2} (See MR2326947.) Is there an equivariant continuous map (called Cannon--Thurston map) from the unit circle

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