Contracting Boundary of a Cusped Space

TL;DR

This paper extends the understanding of the contracting boundary in finitely generated groups, showing that if the cusped space has a compact contracting boundary, then the group is relatively hyperbolic with respect to certain subgroups.

Contribution

It proves that a finitely generated group is relatively hyperbolic if its associated cusped space has a contracting, compact boundary, generalizing previous results on hyperbolicity.

Findings

Cusped space with contracting horoballs has compact contracting boundary

Group is relatively hyperbolic if the cusped space's boundary is compact

Results extend known hyperbolicity criteria to relative hyperbolicity

Abstract

Let $G$ be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics is compact then $G$ is a hyperbolic group. Let $\mathcal{H}$ be a finite collection of finitely generated infinite index subgroups of $G$. Let $G^h$ be the cusped space obtained by attaching combinatorial horoballs to each left cosets of elements of $\mathcal {H}$. In this article, we prove that if the combinatorial horoballs are contracting and $G^h$ has compact contracting boundary then $G$ is hyperbolic relative to $\mathcal{H}$.