Geodesic
Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups
Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2115-2137

Voir la notice de l'article provenant de la source EMS Press

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.
DOI : 10.4171/jems/417
Classification : 20-XX, 22-XX, 30-XX, 57-XX
Keywords: Floyd boundary, convergence actions, quasi-isometric maps 
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     author = {Victor Gerasimov and Leonid Potyagailo},
     title = {Quasi-isometric maps and {Floyd} boundaries of relatively hyperbolic groups},
     journal = {Journal of the European Mathematical Society},
     pages = {2115--2137},
     publisher = {mathdoc},
     volume = {15},
     number = {6},
     year = {2013},
     doi = {10.4171/jems/417},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/417/}
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Victor Gerasimov; Leonid Potyagailo. Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups. Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2115-2137. doi: 10.4171/jems/417

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