Generic free subgroups and statistical hyperbolicity

This paper studies the generic behavior of k -tuples of elements for k \geq 2 in a proper group action with contracting elements, with applications toward relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of k elements for any fixed k \geq 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 statistically convex-cocompact actions are statistically hyperbolic in the 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.