A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

TL;DR

This paper characterizes relatively hyperbolic groups using boundary theory, showing that certain boundaries are non-empty and compact if and only if the group is relatively hyperbolic with respect to a collection of subgroups.

Contribution

It provides a boundary-theoretic criterion for relative hyperbolicity, connecting Morse and contracting boundaries with the group's algebraic structure.

Findings

Non-empty and compact Morse boundary characterizes relative hyperbolicity.

Non-empty and compact contracting boundary characterizes relative hyperbolicity.

Provides an if-and-only-if condition linking boundaries to group properties.

Abstract

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated cusped space. We prove that the pair $(G,\mathcal{H})$ is non-elementary relatively hyperbolic if and only if the Morse boundary $\partial_M^{\mathcal{DL}} G^h$ or the contracting boundary $\partial_c^{\mathcal{FQ}} G^h$ is non-empty and compact.