Rigidity for quasi-Möbius actions on fractal metric spaces

Abstract

In [4], M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(−1)-spaces that can be seen as a metric analog to the "entropy rigidity" theorems of U. Hamenstädt [16] and M. Bourdon [8]. Building on the ideas developed in [4], we establish a rigidity theorem for certain expanding quasi-Möbius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy rigidity result in the coarse geometric setting.

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.…”

Rigidity for convex-cocompact actions on rank-one symmetric spaces

“…In the case that X and Y are bounded and η is linear, each η-quasi-Möbius map f : X → Y is actually bi-Lipschitz [35,Remark 3.2]. In particular, every Möbius map is bi-Lipschitz.…”

“…Any two such metrics are quasisymmetrically equivalent. Moreover, the action of Γ on itself by left multiplication extends naturally to a boundary action Γ ∂Γ that is uniformly quasi-Möbius (with linear distortion function η(t)) with respect to any visual Gromov metric [35,Section 6].…”

“…More recently, there has been interest in knowing whether similar types of rigidity phenomena occur when Γ is not assumed, a priori, to be a lattice [6,35]. The first difficulty encountered is to determine a natural lower bound for Hdim(Λ(Γ)).…”

Invertible Carnot Groups

“…Note that (non-abelian) Carnot groups are fractal in the sense that their topological and Hausdorff dimensions do not agree. When we restrict ourselves to non-fractal metric spaces, we can use recent work of Kinneburg (see [Kin15]) to obtain the following result.…”