Poisson-Furstenberg boundary and growth of groups

Details Poisson-Furstenberg boundary and growth of groups Poisson-Furstenberg boundary and growth of groups

2011-07-27

arxiv.org/abs/1107.5499v1

Mathematics

Group Theory

24 pages

Scientific paper

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.

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