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 $\mathcal{ H}$.
Categories: math.GR math.GT