Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Gerasimov, 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 -