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

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 $Out\left(F_N\right)$, each time under a finite second moment condition on the measure (either with respect to the Teichmüller 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 $Out\left(F_N\right)$, 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 Teichmüller space of the surface or to the Culler–Vogtmann outer space.