# Generic free subgroups and statistical hyperbolicity

## TL;DR

This paper demonstrates that in certain group actions, a generic set of elements typically generates free subgroups and exhibits statistical hyperbolicity, with applications to various classes of groups.

## Contribution

It introduces new results on the generic behavior of elements in group actions with contracting elements, establishing conditions for free subgroup generation and hyperbolic properties.

## Key findings

- Exponential generic sets generate free subgroups of rank k.
- Statistically convex-cocompact actions are statistically hyperbolic.
- Results apply to relatively hyperbolic, CAT(0), and mapping class groups.

## Abstract

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ 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 convex-cocompact action, we show that an exponential generic set of $k$ elements for any fixed $k\ge 2$ 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. Leli\`evre, and C. Mooney. For any proper action with a contracting element, if it satisfies a condition introduced by Dal'bo-Otal-Peign\'e and has purely exponential growth, we obtain the same results on generic free subgroups and statistical hyperbolicity.