A positive proportion of elements of mapping class groups is pseudo-Anosov

Abstract

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.

6 pages

Cited by in corpus (4)

• Problems, Questions, and Conjectures about Mapping Class Groups
• Pseudo-Anosovs are exponentially generic in mapping class groups
• Genericity of contracting elements in groups
• Counting pseudo-Anosovs as weakly contracting isometries