Spectral theorems for random walks on mapping class groups and $\text{Out}(F_N)$

TL;DR

This paper establishes spectral theorems linking the asymptotic growth rates of random walks on mapping class groups and Out(F_N) to Lyapunov exponents, using contraction properties of geodesics and complexes.

Contribution

It provides new spectral theorems connecting random walk asymptotics to Lyapunov exponents for both mapping class groups and Out(F_N), with a general lifting criterion.

Findings

Growth rate of the stretching factor matches Lyapunov exponent.

Drift in the curve complex equals linear growth of translation lengths.

Lifting argument applies to Teichmüller and outer spaces.

Abstract

We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the mapping class group case, we first observe that the drift of the random walk in the curve complex is also equal to the linear growth rate of the translation lengths in this complex. By using a contraction property of typical Teichm\"uller geodesics, we then lift the above fact to the realization of the random walk on the Teichm\"uller space. For the case of $\text{Out}(F_N)$, we follow the same procedure with the free factor complex in place of the curve complex, and the outer space in place of the Teichm\"uller space. A general criterion is given for making the lifting argument possible.