The Poisson boundary of Out(FN)

Abstract

Let $\mu$ be a probability measure on $Out\left(F_N\right)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $Out\left(F_N\right)$. We show that almost every sample path of the random walk on $(Out(F_N),\mu )$, when realized in Culler and Vogtmann’s outer space, converges to the simplex of a free, arational tree. We then prove that the space $\mathcal{FI}$ of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of $(Out(F_N),\mu )$. Using Bestvina and Reynolds’s and Hamenstädt’s description of the Gromov boundary of the complex ${\mathcal{FF}}_N$ of free factors of $F_N$, this gives a new proof of the fact, due to Calegari and Maher, that the realization in ${\mathcal{FF}}_{\mathcal{N}}$ of almost every sample path of the random walk converges to a boundary point. We get in addition that $\partial {\mathcal{FF}}_N$, equipped with the hitting measure, is the Poisson boundary of $(Out(F_N),\mu )$.