Exponential decay in the mapping class group

TL;DR

This paper proves that the likelihood of a random walk in the mapping class group producing non-pseudo-Anosov elements decreases exponentially with walk length, highlighting the rarity of such elements in long random walks.

Contribution

It establishes exponential decay rates for the probability of non-pseudo-Anosov elements in random walks on the mapping class group, extending to sets with bounded translation lengths.

Findings

Probability of non-pseudo-Anosov elements decays exponentially

Random walks tend to produce pseudo-Anosov elements over time

Bounded translation length sets also exhibit exponential decay in probability

Abstract

We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their translation lengths on the complex of curves, then the probability that a random walk lies in R decays exponentially in the length of the random walk.