Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Abstract

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 $\varphi$ into a relatively hyperbolic group $G$ then $H$ is itself relatively hyperbolic with respect to a system of subgroups whose image under $\varphi$ 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 $\varphi$ is an $\alpha$-isometric map for any polynomial distortion function $\alpha.$ As an application of our method we provide in the Appendix a new short proof of a basic theorem of Bowditch characterizing hyperbolicity.