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 admits a quasi-isometric map into a relatively hyperbolic group then is itself relatively hyperbolic with respect to a system of subgroups whose image under is situated within a uniformly bounded distance from the right cosets of the parabolic subgroups of . We then generalize the latter result to the case when is an -isometric map for any polynomial distortion function As an application of our method we provide in the Appendix a new short proof of a basic theorem of Bowditch characterizing hyperbolicity.
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Leonid Potyagailo, Victor Gerasimov, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2115–2137DOI 10.4171/JEMS/417