Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups

Abstract

We study combinatorial modulus on self-similar metric spaces. We give new examples of hyperbolic groups whose boundaries satisfy a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups. We also establish some connections with ℓ p -cohomology.

Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces

“…The introduction of quasimöbius maps has provided a handy tool when studying the quasisymmetric maps and the quasiconformal maps. Many references related to the relationships among quasimöbius maps, quasisymmetric maps and quasiconformal maps have been in literature; see [1,5,6,7,13,15,16,17,18,19,20,21,25,29,30] etc. The precise definition for quasimöbius maps is as follows.…”

On the conformal gauge of a compact metric space

“…In Theorem 3.12, we give metric conditions on X that allow us to compute its AR conformal dimension using another critical exponent Q X , defined from "genuine" curves of X. In Corollary 3.14, we give a proof of the result of Keith and Kleiner mentioned earlier (Remark 1 after Corollary 3.13 in [BouK09]), i.e. when X is approximately self-similar, it suffices to work with the modulus of curves with definite diameter.…”

“…It is defined using coverings of X; therefore, it depends only on the combinatorics of such coverings. In [BouK09], the authors proved several important properties of combinatorial modulus for approximately self-similar sets. By defining a combinatorial modulus of a metric space (X, d) that takes into account all the "annuli" of the space, with some fixed radius ratio, we extend to a more general setting some of these properties.…”

“…In [BouK09] it is stated and used in the self-similar case, see the Remark 1 after Corollary 3.13 therein. This was part of the motivation for working on these questions and they led me to a proof of Theorem 1.2 in this general setting.…”

“…In Section 3.3, inspired by [BouK09], we show that the combinatorial modulus satisfies some kind of sub-multiplicative inequality giving the positiveness of the combinatorial modulus at the critical exponent (Corollary 3.9). We adapt to our situation arguments from [KL04] and [Haïss09] to bound from above the combinatorial modulus by the analytical moduli defined in the tangent spaces.…”