On genericity of pseudo-Anosovs in the Torelli group

TL;DR

This paper proves that in the Torelli group of a closed surface, the likelihood of randomly chosen words not being pseudo-Anosov decreases exponentially as the word length increases.

Contribution

It establishes an exponential decay rate for the probability that a random word in the Torelli group is not pseudo-Anosov, demonstrating genericity of pseudo-Anosov elements.

Findings

Probability of non-pseudo-Anosov words decays exponentially

Pseudo-Anosov elements are generic in the Torelli group

Results hold for any symmetric finite generating set

Abstract

We show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in terms of the length of the word.