Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group H admits a quasi-isometric map φ into a relatively hyperbolic group G then H is itself relatively hyperbolic with respect to a system of subgroups whose image under φ is situated within a uniformly bounded distance from the right cosets of the parabolic subgroups of G . We then generalize the latter result to the case when φ is an α -isometric map for any polynomial distortion function α. As an application of our method we provide in the Appendix a new short proof of a basic theorem of Bowditch characterizing hyperbolicity.