Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Abstract

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\varphi$ into a relatively hyperbolic group $G$ is relatively hyperbolic with respect to a system of subgroups whose image under $\varphi$ is situated in a uniformly bounded distance from the parabolic subgroups of $G$.

17 pages, 1 figure

References in corpus (3)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups
• Floyd maps to the boundaries of relatively hyperbolic groups

Cited by in corpus (21)

• Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups
• Growth tightness for groups with contracting elements
• Relations between various boundaries of relatively hyperbolic groups
• Patterson-Sullivan measures and growth of relatively hyperbolic groups
• Floyd maps to the boundaries of relatively hyperbolic groups
• Boundaries of Dehn fillings
• Blowing up and down compacta with geometrically finite convergence actions of a group
• Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups
• Hausdorff dimension of boundaries of relatively hyperbolic groups
• Controlled Floyd Separation and Non Relatively Hyperbolic Groups
• Similar relatively hyperbolic actions of a group
• Conical limit points and the Cannon-Thurston map
• The Bowditch boundary of $(G,\mathcal{H})$ when $G$ is hyperbolic
• Cannon-Thurston fibers for iwip automorphisms of $F_N$
• Maps between relatively hyperbolic spaces and between their boundaries
• Relatively dominated representations from eigenvalue gaps and limit maps
• Marked length spectrum rigidity in groups with contracting elements
• Random walks and contracting elements I: Deviation inequality and Limit laws
• Boundaries of ${\mathbb{Z}}^n$-free groups
• Equivariant blowups of bounded parabolic points
• Glueing spaces without identifying points