Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

  • Leonid Potyagailo

    Université Lille I, Villeneuve d'Ascq, France
  • Victor Gerasimov

    Universidade Federal de Minas Gerais, Belo Horizonte, Brazil


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 HH admits a quasi-isometric map φ\varphi into a relatively hyperbolic group GG then HH is itself relatively hyperbolic with respect to a system of subgroups whose image under φ\varphi is situated within a uniformly bounded distance from the right cosets of the parabolic subgroups of GG. We then generalize the latter result to the case when φ\varphi is an α\alpha-isometric map for any polynomial distortion function α.\alpha. As an application of our method we provide in the Appendix a new short proof of a basic theorem of Bowditch characterizing hyperbolicity.

Cite this article

Leonid Potyagailo, Victor Gerasimov, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2115–2137

DOI 10.4171/JEMS/417