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October 2003 Boundaries and random walks on finitely generated infinite groups
Anders Karlsson
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Ark. Mat. 41(2): 295-306 (October 2003). DOI: 10.1007/BF02390817

Abstract

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

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Anders Karlsson. "Boundaries and random walks on finitely generated infinite groups." Ark. Mat. 41 (2) 295 - 306, October 2003. https://doi.org/10.1007/BF02390817

Information

Received: 17 December 2001; Published: October 2003
First available in Project Euclid: 31 January 2017

MathSciNet: MR2011923
Digital Object Identifier: 10.1007/BF02390817

Rights: 2003 © Institut Mittag-Leffler

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Vol.41 • No. 2 • October 2003
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