Hyperfiniteness of boundary actions of relatively hyperbolic groups

TL;DR

This paper proves that for relatively hyperbolic groups, the natural boundary actions induce hyperfinite equivalence relations, extending previous results and utilizing recent advances in the field.

Contribution

It establishes hyperfiniteness of boundary actions for relatively hyperbolic groups, generalizing prior results to broader classes of subgroups.

Findings

Boundary actions induce hyperfinite equivalence relations.

Extends Ozawa's result to non-amenable subgroups.

Utilizes recent work of Marquis and Sabok.

Abstract

We show that if $G$ is a finitely generated group hyperbolic relative to a finite collection of subgroups $\mathcal{P}$, then the natural action of $G$ on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite equivalence relation. As a corollary of this, we obtain that the natural action of $G$ on its Bowditch boundary $\partial (G,\mathcal{P})$ also induces a hyperfinite equivalence relation. This strengthens a result of Ozawa obtained for $\mathcal{P}$ consisting of amenable subgroups and uses a recent work of Marquis and Sabok.