The Poisson boundary of $\text{Out}(F_N)$

TL;DR

This paper identifies the Poisson boundary of random walks on Out(F_N) with the space of free, arational trees, showing convergence of sample paths in outer space and providing new proofs of boundary convergence results.

Contribution

It proves that the space of free, arational trees forms the Poisson boundary for Out(F_N) random walks, linking boundary behavior with geometric structures in outer space.

Findings

Sample paths converge to the simplex of free, arational trees in outer space.

The space of free, arational trees with the hitting measure is the Poisson boundary.

Convergence in the complex of free factors aligns with boundary points, confirming known results.

Abstract

Let $\mu$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost every sample path of the random walk on $(\text{Out}(F_N),\mu)$, 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 $(\text{Out}(F_N),\mu)$. Using Bestvina-Reynolds' and Hamenst\"adt'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 $\mathcal{FF_N}$ 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 $(\text{Out}(F_N),\mu)$.