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Groups, Geometry, and Dynamics

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Volume 15, Issue 1, 2021, pp. 101–140
DOI: 10.4171/GGD/593

Published online: 2020-12-24

Generic free subgroups and statistical hyperbolicity

Suzhen Han[1] and Wen-Yuan Yang[2]

(1) Peking University, Beijing, China
(2) Peking University, Beijing, China

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.

Keywords: Contracting elements, free subgroups, growth rate, statistical hyperbolicity, genericity

Han Suzhen, Yang Wen-Yuan: Generic free subgroups and statistical hyperbolicity. Groups Geom. Dyn. 15 (2021), 101-140. doi: 10.4171/GGD/593