Genericity of contracting elements in groups

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 ) 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.