Generic free subgroups and statistical hyperbolicity (Q2035110)

Summary: 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.