We establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. We obtain as corollaries results on the exponential genericity for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 elements in CAT(0) groups, and the set of pseudo-Anosov elements in mapping class groups. For a proper action with purely exponential growth, we show that the set of contracting elements is generic. In particular, for mapping class groups, the set of pseudo-Anosov elements is generic in a sufficiently large subgroup, provided that the subgroup has purely exponential growth. By Roblin’s work, we obtain that the set of hyperbolic elements is generic in any discrete group action on CAT( $$-1$$ - 1 ) space with finite BMS measure. We present applications to the number of conjugacy classes of non-rank-1 elements in CAT(0) groups with rank-1 elements.

中文翻译：

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组内承包要素的通用性

我们确定，对于统计上的凸协紧动作，收缩元素在计数度量中呈指数通用。我们获得了相对双曲群中的双曲元素集、CAT(0) 群中的 rank-1 元素集以及映射类群中的伪 Anosov 元素集的指数通用性结果作为推论。对于纯指数增长的适当动作，我们表明收缩元素集是通用的。特别是，对于映射类群，伪 Anosov 元素集在足够大的子群中是通用的，前提是该子群具有纯指数增长。通过 Roblin 的工作，我们得到双曲元素的集合在 CAT( $$-1$$ - 1 ) 空间上具有有限 BMS 测度的任何离散群动作中是通用的。