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

Authors: Donald I. Cartwright and P. M. Soardi
Journal: Proc. Amer. Math. Soc. 107 (1989), 817-823
MSC: Primary 60J50; Secondary 05C05, 43A05, 60J15
MathSciNet review: 984784
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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].

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Keywords: Random walks, free groups, ends of trees, boundaries of groups
Article copyright: © Copyright 1989 American Mathematical Society

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