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Suppose G is a finitely generated group and H is a subgroup of G. Let ₂^FQG 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 ₂^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 BinSun, if the limit set Λ (H) is compact and the action of H on ₂^FQG is non-elementary then H becomes an acylindrically hyperbolic group
Pal et al. (Sat,) studied this question.
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