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

Let μ be a probability on a free group Γof rank r ≥ 2. Assume that Supp(μ) is notcontained in a cyclic subgroup of Γ. We show that if (Xn)n≥0 is the right random walk on Γ determined by μ, then with probability 1, Xn converges (in the natural sense) to an infinite reduced word. The spaceΩ of infinite reduced words carries a unique probability v such that (Ω, v) is a frontier of (Γ,μ) in the sense of Furstenberg [10]. This result extends to the right random walk (Xn) determined by a probability p. on the group G of automorphisms of an arbitrary infinite locally finite tree T. Assuming that Supp(μ) is not contained in any amenable closed subgroup of G, then with probability 1 there is an end ω ofT suchthat Xnv converges to ωfor each v ∈ T. Our methods are principally drawn from [9] and [10]