Pseudo-Anosovs are exponentially generic in mapping class groups

TL;DR

This paper proves that pseudo-Anosov elements are exponentially prevalent in the mapping class group, meaning non-pseudo-Anosov elements become negligible as word length increases, using a strategy applicable to weakly hyperbolic groups.

Contribution

It demonstrates that pseudo-Anosov elements are exponentially generic in the mapping class group, extending the understanding of their distribution without relying on automatic structures.

Findings

Proportion of non-pseudo-Anosov elements decreases exponentially with word length.

Any finite subset can be extended to a generating set with exponential decay of non-pseudo-Anosov elements.

Strategy applies broadly to weakly hyperbolic groups, not just mapping class groups.

Abstract

Given a finite generating set $S$, let us endow the mapping class group of a closed hyperbolic surface with the word metric for $S$. We discuss the following question: does the proportion of non-pseudo-Anosov mapping classes in the ball of radius $R$ decrease to 0 as $R$ increases? We show that any finite subset $S'$ of the mapping class group is contained in a finite generating set $S$ such that this proportion decreases exponentially. Our strategy applies to weakly hyperbolic groups and does not refer to the automatic structure of the group.