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.

中文翻译：

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泛型自由子群与统计双曲

本文研究了在适当的带有收缩元素的组动作中，$ k \ geq 2 $的$ k $元组的一般行为，并应用于相对双曲型组，CAT（0）组和映射类组。对于一类统计上凸共紧缩作用，我们证明了任何固定的$ k \ geq 2 $的指数k $$元素通用集都会生成一个准等距嵌入的自由子组，其秩为$ k $。对于$ k = 2 $，我们研究了群体动作的蔓延性质，并建立了在M. Duchin，S。Lelièvre和C. Mooney的意义上，统计上凸的共紧缩动作在统计上是双曲线的。

对于任何带有收缩元素的适当动作，如果它满足Dal'bo-Otal-Peigné引入的条件并具有纯粹的指数增长，则我们在仿制药自由子群和统计双曲性上可获得相同的结果。