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

Abstract

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