New York Journal of Mathematics
Volume 26 (2020), 1213-1231


Abhijit Pal and Rahul Pandey

Acylindrical hyperbolicity of subgroups

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Published: October 17, 2020.
Keywords: Contracting boundary, Morse boundary, contracting quasi-geodesic, acylindrically hyperbolic group.
Subject: 20F65, 20F67, 57M07.

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 [7]. 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 [24], 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.


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.

Author information

Abhijit Pal:
Indian Institute of technology
Kanpur, India


Rahul Pandey:
Indian Institute of technology
Kanpur, India