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]

Cartwright, D., Soardi, P. (1989). Convergence to ends for random walks on the automorphism group of a tree. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 107(3), 817-823 [10.2307/2048184].

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

Soardi, PM
1989

Abstract

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]
Articolo in rivista - Articolo scientifico
Random walks; free groups; boundaries of groups; amenable subgroup
English
1989
107
3
817
823
none
Cartwright, D., Soardi, P. (1989). Convergence to ends for random walks on the automorphism group of a tree. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 107(3), 817-823 [10.2307/2048184].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/18346
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