Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

TL;DR

This paper investigates the relationship between Floyd and Bowditch boundaries of relatively hyperbolic groups, and characterizes groups quasi-isometrically embedded into such groups as relatively hyperbolic with controlled subgroups.

Contribution

It describes the kernel of the canonical boundary map and establishes conditions under which a group quasi-isometrically maps into a relatively hyperbolic group is itself relatively hyperbolic.

Findings

Kernel of the boundary map characterized

Quasi-isometric images are relatively hyperbolic with bounded subgroups

Provides new insights into boundary maps and subgroup structures

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 $\phi$ into a relatively hyperbolic group $G$ is relatively hyperbolic with respect to a system of subgroups whose image under $\phi$ is situated in a uniformly bounded distance from the parabolic subgroups of $G$.