Abstract
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.
Highlights
The conformal dimension of a metric space, introduced by Pansu, is the infimal Hausdorff dimension of all the quasisymmetrically equivalent metrics on the space
The initial motivation for the introduction of this invariant by Pansu in [35] was in the study of the large scale geometry of negatively curved homogeneous spaces, for which the conformal dimension can be computed explicitly
In general it is an invariant that is very hard to compute. It has found applications in other areas of geometric group theory and dynamical systems. These include the work of Bonk and Kleiner on the rigidity of quasi-Möbius group actions [2]; the works of Bonk and Kleiner [3] and Haïssinsky [21] on Cannon’s conjecture and the boundary characterisation of Kleinian groups; the works of Haïssinsky and Pilgrim on the characterisation of rational maps among coarse expanding conformal dynamical systems on the 2-sphere [23]; the works of Bourdon and Kleiner focussing on the relations between the p-cohomology, the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property [5,6]; and the works of the second author on conformal dimension bounds for small cancellation and random groups [31,32], as well as further connections to actions on L p-spaces [8,16]
Summary
The conformal dimension of a metric space, introduced by Pansu, is the infimal Hausdorff dimension of all the quasisymmetrically equivalent metrics on the space. Confdim ∂∞G = max{Confdim ∂∞Gi : Gi infinite}, where max ∅ = 0 In light of this result, Theorem 1.1 reduces to the following: Theorem 1.4 Suppose G is a hyperbolic group with a graph of groups decomposition of G with vertex groups {Gi } and all edge groups 2-ended, if G is not virtually free, Confdim ∂∞G = max {1} ∪ {Confdim ∂∞Gi }. Proof of Corollary 1.2 Suppose G admits a finite hierarchy of graph of groups decompositions over finite and 2-ended subgroups, ending with vertex groups that are elementary or virtually Fuchsian; such groups have conformal dimension at most 1. Remark 1.6 The groups considered in Corollary 1.2, when torsion free, are the groups Wise suggests might be the hyperbolic virtual limit groups [42, Section 1.4]
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