Counting pseudo-Anosovs as weakly contracting isometries

TL;DR

This paper demonstrates that pseudo-Anosov elements are generically present as weakly contracting isometries in various groups, extending the concept to hierarchically hyperbolic groups and establishing a quasi-isometry invariant counting theory.

Contribution

It introduces a new genericity result for pseudo-Anosov elements across different group classes, including hierarchically hyperbolic groups, and develops a quasi-isometry invariant framework.

Findings

Pseudo-Anosov elements are generic in Cayley graphs of mapping class groups.

Analogous genericity results hold for rank-one CAT(0) groups.

Morse elements are also generic in groups quasi-isometric to hierarchically hyperbolic groups.

Abstract

We show that pseudo-Anosov mapping classes are generic in every Cayley graph of the mapping class group of a finite-type hyperbolic surface. Our method also yields an analogous result for rank-one CAT(0) groups and hierarchically hyperbolic groups with Morse elements. Finally, we prove that Morse elements are generic in every Cayley graph of groups that are quasi-isometric to (well-behaved) hierarchically hyperbolic groups. This gives a quasi-isometry invariant theory of counting group elements in groups beyond relatively hyperbolic groups.