JournalsggdVol. 15, No. 1pp. 101–140

Generic free subgroups and statistical hyperbolicity

  • Suzhen Han

    Peking University, Beijing, China
  • Wen-Yuan Yang

    Peking University, Beijing, China
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Abstract

This paper studies the generic behavior of kk-tuples of elements for k2k \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 kk elements for any fixed k2k \geq 2 generates a quasi-isometrically embedded free subgroup of rank kk. For k=2k = 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.

Cite this article

Suzhen Han, Wen-Yuan Yang, Generic free subgroups and statistical hyperbolicity. Groups Geom. Dyn. 15 (2021), no. 1, pp. 101–140

DOI 10.4171/GGD/593