Hyperfiniteness of boundary actions of relatively hyperbolic groups

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.