Acylindrical Hyperbolicity of Subgroups

Suppose $G$ is a finitely generated group and $H$ is a subgroup of $G$. Let ${\mathrm{\partial }}_c^{\mathcal{F}\mathcal{Q}}G$ denote the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay \cite{cashen2017}. In this article, we show that if the limit set $\mathrm{\Lambda }(H)$ of $H$ in ${\mathrm{\partial }}_c^{\mathcal{F}\mathcal{Q}}G$ is compact and contains at least three points then the action of the subgroup $H$ on the space of distinct triples ${\mathrm{\Theta }}_3(\mathrm{\Lambda }(H))$ is properly discontinuous. By applying a result of B. Sun \cite{BinSun}, if the limit set $\mathrm{\Lambda }(H)$ is compact and the action of $H$ on ${\mathrm{\partial }}_c^{\mathcal{F}\mathcal{Q}}G$ is non-elementary then $H$ becomes an acylindrically hyperbolic group