9 Problems related to Cannon's Conjecture
Problem 9.1 (Comments) [Cannon's Conjecture, Version I] If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a (Gromov) hyperbolic group with
dvipng error! exitcode was 2 (signal 0), transscript follows:homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:, then
dvipng error! exitcode was 2 (signal 0), transscript follows:acts geometrically on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 9.2 (Comments) [Cannon's Conjecture, Version II] Under the same assumptions on
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:
Remark Perelman's proof of Thurston's geometrization conjecture implies that the Cannon's conjecture is equivalent to the finding that such
dvipng error! exitcode was 2 (signal 0), transscript follows:is commensurable to a 3-manifold group: There exists an exact sequence
dvipng error! exitcode was 2 (signal 0), transscript follows:
with
dvipng error! exitcode was 2 (signal 0), transscript follows:finite and
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark If
dvipng error! exitcode was 2 (signal 0), transscript follows:is hyperbolic and torsion-free then
dvipng error! exitcode was 2 (signal 0), transscript follows:iff
dvipng error! exitcode was 2 (signal 0), transscript follows:is a
dvipng error! exitcode was 2 (signal 0), transscript follows:group (a 3-dimensional Poincar\'{e} duality group), see MR1096169.
Problem 9.3 (Comments) [Conjecture of C.T.C. Wall] Every
dvipng error! exitcode was 2 (signal 0), transscript follows:group is a 3-manifold group.
Problem 9.4 (Comments) [Cannon's Conjecture, Relative Version I] If
dvipng error! exitcode was 2 (signal 0), transscript follows:is hyperbolic and
dvipng error! exitcode was 2 (signal 0), transscript follows:is homeomorphic to the Sierpinski carpet, then
dvipng error! exitcode was 2 (signal 0), transscript follows:acts geometrically on a convex subset of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark This follows from the Cannon's Conjecture by doubling.
Problem 9.5 (Comments) [Cannon's Conjecture, Relative Version II] For
dvipng error! exitcode was 2 (signal 0), transscript follows:hyperbolic relative to
dvipng error! exitcode was 2 (signal 0), transscript follows:for a collection of virtually-
dvipng error! exitcode was 2 (signal 0), transscript follows:subgroups
dvipng error! exitcode was 2 (signal 0), transscript follows:, if the Bowditch boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:is
dvipng error! exitcode was 2 (signal 0), transscript follows:, then
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:or Sierpinski carpet.
Problem 9.6 (Comments) [Cannon's Conjecture, Analytic Version] If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a hyperbolic group with
dvipng error! exitcode was 2 (signal 0), transscript follows:homeomorphic to the Sierpinski carpet, then the visual metric on
dvipng error! exitcode was 2 (signal 0), transscript follows:is quasisymmetric to some round Sierpinski metric:
dvipng error! exitcode was 2 (signal 0), transscript follows:
Recent work of Mario Bonk gives simplifications and partial answers here.
Problem 9.7 (Comments) Prove Cannon's conjecture under additional assumptions, such as
dvipng error! exitcode was 2 (signal 0), transscript follows:
, in Haken case (without appealing to Thurston's proof of the hyperbolization theorem)dvipng error! exitcode was 2 (signal 0), transscript follows:
advipng error! exitcode was 2 (signal 0), transscript follows:
group that splits over a surface groupdvipng error! exitcode was 2 (signal 0), transscript follows:
acts on a CAT(0) cube complex- ....
Remark Cannon's Conjecture is known for Coxeter groups
dvipng error! exitcode was 2 (signal 0), transscript follows:(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
dvipng error! exitcode was 2 (signal 0), transscript follows:.
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
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:, where
dvipng error! exitcode was 2 (signal 0), transscript follows:. For sufficiently long elements
dvipng error! exitcode was 2 (signal 0), transscript follows:the quotient
dvipng error! exitcode was 2 (signal 0), transscript follows:are known to be hyperbolic.
a. Prove that if
dvipng error! exitcode was 2 (signal 0), transscript follows:'s are sufficiently long then
dvipng error! exitcode was 2 (signal 0), transscript follows:is an (absolute)
dvipng error! exitcode was 2 (signal 0), transscript follows:group, by, say, computing
dvipng error! exitcode was 2 (signal 0), transscript follows:.
b. Assuming that each
dvipng error! exitcode was 2 (signal 0), transscript follows:is a 3-manifold group, show that
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:is a hyperbolic group,
dvipng error! exitcode was 2 (signal 0), transscript follows:is a uniformly quasi-Moebius action on a metric space which is topologically conjugate to the action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on its ideal boundary. Assume that
dvipng error! exitcode was 2 (signal 0), transscript follows:is Ahlfors
dvipng error! exitcode was 2 (signal 0), transscript follows:-regular and has topological dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then
dvipng error! exitcode was 2 (signal 0), transscript follows:is quasi-symmetric to the round
dvipng error! exitcode was 2 (signal 0), transscript follows:-sphere. In particular,
dvipng error! exitcode was 2 (signal 0), transscript follows:acts geometrically on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Therefore, given a hyperbolic group
dvipng error! exitcode was 2 (signal 0), transscript follows:with
dvipng error! exitcode was 2 (signal 0), transscript follows:homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:one would like to replace the visual metric
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:with a quasisymmetrically equivalent one, which has Hausdorff dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:. Since Hausdorff dimension (
dvipng error! exitcode was 2 (signal 0), transscript follows:) of a metric compact homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:is
dvipng error! exitcode was 2 (signal 0), transscript follows:, one could try to minimize Hausdorff dimension in the quasi-conformal gauge of
dvipng error! exitcode was 2 (signal 0), transscript follows:, i.e. the collection
dvipng error! exitcode was 2 (signal 0), transscript follows:of metric spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:which are quasisymmetric to
dvipng error! exitcode was 2 (signal 0), transscript follows:. This motivates the following:
Definition 9.2 For a metric space
dvipng error! exitcode was 2 (signal 0), transscript follows:, define its Pansu conformal dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:
Likewise, Ahlfors regular Pansu conformal dimension of
dvipng error! exitcode was 2 (signal 0), transscript follows:is
dvipng error! exitcode was 2 (signal 0), transscript follows:
The importance of the latter comes from
Theorem 9.3 [M.~Bonk, B.~Kleiner, MR2116315] Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a hyperbolic group,
dvipng error! exitcode was 2 (signal 0), transscript follows:is homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:. If the
dvipng error! exitcode was 2 (signal 0), transscript follows:is attained then
dvipng error! exitcode was 2 (signal 0), transscript follows:acts geometrically on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark Bourdon--Pajot examples MR1979183 show that the
dvipng error! exitcode was 2 (signal 0), transscript follows:for the boundaries of hyperbolic groups is not always attained.
Generally,
dvipng error! exitcode was 2 (signal 0), transscript follows:attained iff there is a Loewner metric in
dvipng error! exitcode was 2 (signal 0), transscript follows:(which is then minimizing).
Problem 9.9 (Comments) [Conjecture of Bruce Kleiner] For a hyperbolic group
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:
where the infimum is taken over all geometric actions of
dvipng error! exitcode was 2 (signal 0), transscript follows:on metric spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:. A bolder conjecture would be that, when the infimum is attained, it is attained by a visual metric.
Problem 9.10 (Comments) What is
dvipng error! exitcode was 2 (signal 0), transscript follows:of the standard Sierpinski carpet? In particular, does the above conjecture hold?
Problem 9.11 (Comments) [Juha Heinonen] Under what assumptions on hyperbolic groups
dvipng error! exitcode was 2 (signal 0), transscript follows:with
dvipng error! exitcode was 2 (signal 0), transscript follows:-Loewner boundary
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:is the support of a measured lamination on a surface
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:consists of topological disks. Lift this lamination to a lamination
dvipng error! exitcode was 2 (signal 0), transscript follows:in the unit disk
dvipng error! exitcode was 2 (signal 0), transscript follows:and define the following equivalence relation
dvipng error! exitcode was 2 (signal 0), transscript follows::
1. The closure of each component of
dvipng error! exitcode was 2 (signal 0), transscript follows:in the closed disk
dvipng error! exitcode was 2 (signal 0), transscript follows:is an equivalence class.
2. If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a geodesic which is not on the boundary of a component of
dvipng error! exitcode was 2 (signal 0), transscript follows:, then the closure of
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:is an equivalence class. The rest of the points of
dvipng error! exitcode was 2 (signal 0), transscript follows:are equivalent only to themselves.
Note that
dvipng error! exitcode was 2 (signal 0), transscript follows:is
dvipng error! exitcode was 2 (signal 0), transscript follows:-invariant and that the equivalence classes are cells. Therefore the quotient
dvipng error! exitcode was 2 (signal 0), transscript follows:is homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:and the group
dvipng error! exitcode was 2 (signal 0), transscript follows:acts on
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:--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
dvipng error! exitcode was 2 (signal 0), transscript follows:. It is not even clear that there exists a Gromov-hyperbolic space
dvipng error! exitcode was 2 (signal 0), transscript follows:with the ideal boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the action
dvipng error! exitcode was 2 (signal 0), transscript follows:extends to a uniformly quasi-isometric quasi-action
dvipng error! exitcode was 2 (signal 0), transscript follows:.
One can reformulate this problem using theory of Kleinian groups as follows. According to the Ending Lamination Conjecture, there exists a discrete embedding
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the ending lamination of
dvipng error! exitcode was 2 (signal 0), transscript follows:is
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 9.13 (Comments) \label{ct1} The limit set of the Kleinian group
dvipng error! exitcode was 2 (signal 0), transscript follows:is locally connected.
In the presence of two geodesic laminations, the limit set of
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:(the ideal boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:as an abstract group) to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Then Problem \ref{ct} is equivalent to \ref{ct2}.
Remark Positive solution of Problem \ref{ct2} is known in certain cases. For instance, Jim Cannon and Bill Thurston showed this for laminations which are stable for a pseudo-Anosov homeomorphism. Yair Minsky MR1257060 proved this under the assumption that the injectivity radius of
dvipng error! exitcode was 2 (signal 0), transscript follows:is bounded away from zero. Curt McMullen proved this in the case when
dvipng error! exitcode was 2 (signal 0), transscript follows:is the fundamental group of once punctured torus of quadruply punctured sphere, MR1859018. A complete solution of this problem is claimed in the recent preprint of Mahan Mj (Mahan Mitra) Mj07.
Problem 9.15 (Comments) [Mahan Mitra] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a hyperbolic subgroup of a hyperbolic group (we do not assume that
dvipng error! exitcode was 2 (signal 0), transscript follows:is quasiconvex). Is it true that there exists an equivariant continuous map
dvipng error! exitcode was 2 (signal 0), transscript follows:
See MR1604882 for partial results in this direction.