Generic free subgroups and statistical hyperbolicity

DOI10.4171/GGD/593arXiv1812.06265MaRDI QIDQ6311195

Suzhen Han, Wen-Yuan Yang

Publication date: 15 December 2018

Abstract: This paper studies the generic behavior of $k$-tuple elements for $kge2$ 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 $kge2$ 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.

Mathematics Subject Classification ID

Geometric group theory (20F65) Hyperbolic groups and nonpositively curved groups (20F67)

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