Subgroup mixing and random walks in groups acting on hyperbolic spaces

TL;DR

This paper investigates the dynamics of acylindrically hyperbolic groups acting on their convex cocompact subgroups, showing that random walks induce highly mixing and transitive actions under certain conditions.

Contribution

It introduces new results on the topological dynamics of group actions on subgroups and extends existing random walk results to this setting.

Findings

Random walks produce elements with strong mixing properties.

The action is highly topologically transitive when no finite normal subgroups exist.

Technical results on convex cocompact subgroups support these conclusions.

Abstract

We study the topological dynamics of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. We show that, for any suitable probability measure $\mu$, random walks with respect to $\mu$ will produce elements with strong mixing properties for this action asymptotically almost surely. In particular, when the group has no finite normal subgroups this implies that the action is highly topologically transitive. Along the way, we prove technical results about convex cocompact subgroups which allow us to extend some results on random walks of Abbott and the first author.