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Poisson-Furstenberg boundary and growth of groups. (English) Zbl 1417.60004

Summary: We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A07 Means on groups, semigroups, etc.; amenable groups
20F65 Geometric group theory
20F69 Asymptotic properties of groups
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