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

## TL;DR

This paper investigates the distribution of pseudo-Anosov elements within the mapping class group, showing that a positive proportion of elements are pseudo-Anosov in large balls of the Cayley graph, supporting a conjecture about their prevalence.

## Contribution

It proves that the proportion of pseudo-Anosov elements remains bounded away from zero in large balls, providing partial evidence for the conjecture that this proportion tends to one.

## Key findings

- Proportion of pseudo-Anosov elements is bounded away from zero.
- Results extend to large classes of subgroups.
- Supports conjecture that pseudo-Anosov elements dominate asymptotically.

## 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.