Convergence to ends for random walks on the automorphism group of a tree

Abstract:

Let $\mu$ be a probability on a free group $\Gamma$ of rank $r \geq 2$. Assume that $\operatorname {Supp} \left ( \mu \right )$ is not contained in a cyclic subgroup of $\Gamma$. We show that if ${\left ( {{X_n}} \right )_{n \geq 0}}$ is the right random walk on $\Gamma$ determined by $\mu$, then with probability 1, ${X_n}$ converges (in the natural sense) to an infinite reduced word. The space $\Omega$ of infinite reduced words carries a unique probability $\nu$ such that $\left ( {\Omega ,\nu } \right )$ is a frontier of $\left ( {\Gamma ,\mu } \right )$ in the sense of Furstenberg [10]. This result extends to the right random walk $\left ( {{X_n}} \right )$ determined by a probability $\mu$ on the group $G$ of automorphisms of an arbitrary infinite locally finite tree $T$. Assuming that $\operatorname {Supp} \left ( \mu \right )$ is not contained in any amenable closed subgroup of $G$, then with probability 1 there is an end $\omega$ of $T$ such that ${X_n}\upsilon$ converges to $\omega$ for each $\upsilon \in T$. Our methods are principally drawn from [9] and [10].