Generic free subgroups and statistical hyperbolicity

Abstract

This paper studies the generic behavior of k-tuple elements for k ≥ 2 in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convexcocompact action, we show that an exponential generic set of k elements for any fixed k ≥ 2 generates a quasi-isometrically embedded free subgroup of rank k. For k = 2, we study the sprawl property of group actions and establish that the class of statistically convex-cocompact actions is statistically hyperbolic in a sense of M. Duchin, S. Lelièvre, and C. Mooney.For any proper action with a contracting element, if it satisfies a condition introduced by Dal'bo-Otal-Peigné and has purely exponential growth, we obtain the same results on generic free subgroups and statistical hyperbolicity.