Some ergodic properties of metrics on hyperbolic groups

TL;DR

This paper investigates ergodic properties of group actions on hyperbolic boundaries using an analogue of geodesic flow, providing new insights into the dynamics of Gromov-hyperbolic groups and their boundary measures.

Contribution

It introduces an analogue of geodesic flow for hyperbolic groups to analyze ergodicity of boundary actions, advancing understanding of hyperbolic group dynamics.

Findings

Established ergodicity properties of boundary actions.

Constructed an analogue of geodesic flow for hyperbolic groups.

Analyzed measure classes on boundary pairs and their dynamics.

Abstract

Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated Patterson-Sullivan measure class $[\nu]$ on $\partial^{(2)}\Gamma$, and its square $[\nu\times\nu]$ on $\partial^{(2)}\Gamma$ -- the space of distinct pairs of points on the boundary. We construct an analogue of a geodesic flow to study ergodicity properties of the $\Gamma$-actions on $(\partial\Gamma,\nu)$ and on $(\partial^{(2)}\Gamma,[\nu\times\nu])$.