Central limit theorems for mapping class groups and $\text{Out}(F_N)$

TL;DR

This paper establishes central limit theorems for random walks on the mapping class group and Out(F_N), describing the statistical spread of geometric quantities under these dynamics.

Contribution

It introduces a general criterion for proving CLTs for the Busemann cocycle on horoboundaries, applied to Teichmüller and outer spaces.

Findings

CLTs hold for mapping class groups and Out(F_N) under finite second moment conditions.

Describes the distribution of lengths of curves and conjugacy classes after random automorphisms.

Provides a unified approach to central limit theorems in geometric group theory contexts.

Abstract

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either with respect to the Teichm\"uller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of $\text{Out}(F_N)$, this describes the spread of the length of primitive conjugacy classes in $F_N$ under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichm\"uller space of the surface, or to Culler--Vogtmann's outer space.