Invariant measures on the space of horofunctions of a word hyperbolic group

TL;DR

This paper studies invariant measures on horofunctions of hyperbolic groups, establishing finiteness and ergodic properties, and applies these to prove a new ergodic theorem for spherical averages in hyperbolic groups.

Contribution

It introduces a natural equivalence relation on horofunctions, proves finiteness of ergodic invariant measures, and applies these results to ergodic theorems for hyperbolic group actions.

Findings

Finitely many ergodic invariant measures under the relation.

Invariant measures lead to virtual ergodicity in product spaces.

New ergodic theorem for spherical averages in hyperbolic groups.

Abstract

We introduce a natural equivalence relation on the space $\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if $\eta$ is such a measure and $G$ acts on a space $(X,\mu)$ by p.m.p. transformations then $\eta \times \mu$ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.