Acylindrical Hyperbolicity of Subgroups

Suppose G is a finitely generated group and H is a subgroup of G. Let ∂cFQG denote the contracting boundary of G with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay {cashen2017}. In this article, we show that if the limit set $Λ(H)$ of H in ∂cFQG is compact and contains at least three points then the action of the subgroup H on the space of distinct triples Θ₃(Λ(H)) is properly discontinuous. By applying a result of B. Sun {BinSun}, if the limit set $Λ(H)$ is compact and the action of H on ∂cFQG is non-elementary then H becomes an acylindrically hyperbolic group