# Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Victor Gerasimov; Leonid Potyagailo

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2115-2137
- ISSN: 1435-9855

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topGerasimov, Victor, and Potyagailo, Leonid. "Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups." Journal of the European Mathematical Society 015.6 (2013): 2115-2137. <http://eudml.org/doc/277801>.

@article{Gerasimov2013,

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.},

author = {Gerasimov, Victor, Potyagailo, Leonid},

journal = {Journal of the European Mathematical Society},

keywords = {Floyd boundary; convergence actions; quasi-isometric maps; relatively hyperbolic groups; finitely generated groups; actions by homeomorphisms; relatively hyperbolic groups; Floyd boundaries; convergence actions; quasi-isometric maps; finitely generated groups; actions by homeomorphisms},

language = {eng},

number = {6},

pages = {2115-2137},

publisher = {European Mathematical Society Publishing House},

title = {Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups},

url = {http://eudml.org/doc/277801},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Gerasimov, Victor

AU - Potyagailo, Leonid

TI - Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 6

SP - 2115

EP - 2137

AB - 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.

LA - eng

KW - Floyd boundary; convergence actions; quasi-isometric maps; relatively hyperbolic groups; finitely generated groups; actions by homeomorphisms; relatively hyperbolic groups; Floyd boundaries; convergence actions; quasi-isometric maps; finitely generated groups; actions by homeomorphisms

UR - http://eudml.org/doc/277801

ER -

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