Hyperfiniteness of boundary actions of relatively hyperbolic groups

We show that if G is a finitely generated group hyperbolic relative to a finite collection of subgroups 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 ∂ (G,P) also induces a hyperfinite equivalence relation. This strengthens a result of Ozawa obtained for P consisting of amenable subgroups and uses a recent work of Marquis and Sabok.