Counting pseudo-Anosovs as weakly contracting isometries

Abstract

Let $S$ be a finite generating set of the mapping class group of a finite-type hyperbolic surface. We show that mapping classes supported on a fixed subsurface are not generic in the word metric with respect to $S$. We also show that pseudo-Anosov mapping classes are generic in the word metric with respect to $S'$, where $S'$ is $S$ plus a single mapping class. We also observe the analogous results for well-behaved hierarchically hyperbolic groups and groups quasi-isometric to them. This gives a version of quasi-isometry invariant theory of counting group elements in groups.