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.