Suppose G is a finitely generated group and H is a subgroup of G. Let ∂_c^FQG denote the contracting boundary of G with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay [7]. In this article, we show that if the limit set Λ(H) of H in ∂_c^FQG 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 [24], if the limit set Λ(H) is compact and the action of H on ∂_c^FQG is non-elementary then H becomes an acylindrically hyperbolic group.

We thank the anonymous referee for his/her valuable comments and suggestions which has helped in improving the exposition of this article from an earlier draft. Research of the first author was supported by DST-INSPIRE Grant IFA12-MA-19.