Maps between relatively hyperbolic spaces and between their boundaries

TL;DR

This paper explores the relationship between maps of relatively hyperbolic groups and quasisymmetric boundary embeddings, establishing correspondences that characterize groups with polynomial distortion embeddings into hyperbolic spaces.

Contribution

It introduces a correspondence between quasi-isometric embeddings respecting peripherals and boundary quasisymmetric embeddings, extending previous results to relatively hyperbolic groups.

Findings

Characterizes groups hyperbolic relative to virtually nilpotent subgroups via polynomial distortion embeddings.

Establishes a correspondence between group embeddings and boundary quasisymmetric maps.

Generalizes Bonk and Schramm's result to relatively hyperbolic groups.

Abstract

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.