Contracting boundary of a cusped space

Abstract

Let [Formula: see text] be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of [Formula: see text] with the topology of fellow traveling quasi-geodesics is compact then [Formula: see text] is a hyperbolic group. Let [Formula: see text] be a finite collection of finitely generated infinite index subgroups of [Formula: see text]. Let [Formula: see text] be the cusped space obtained by attaching combinatorial horoballs to each left coset of elements of [Formula: see text]. In this article, we prove that if the combinatorial horoballs are contracting and [Formula: see text] has compact contracting boundary then [Formula: see text] is hyperbolic relative to [Formula: see text].