Topological flows for hyperbolic groups

TL;DR

This paper studies the boundary measures of hyperbolic groups, revealing a unique invariant measure class with maximal Hausdorff dimension and applying this to characterize hyperbolic metrics through mean distortion.

Contribution

It introduces a unique invariant measure class for hyperbolic groups and connects boundary measure properties to hyperbolic metric characterization.

Findings

Existence of a unique group-invariant Radon measure class with maximal Hausdorff dimension

Characterization of hyperbolic metrics via mean distortion

Application to boundary measure theory in hyperbolic groups

Abstract

We show that for every non-elementary hyperbolic group the Bowen-Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.