Poisson–Furstenberg boundary of random walks on wreath products and free metabelian groups

Abstract

We study the Poisson–Furstenberg boundary of random walks on C = A ≀ B, where A = ℤd and B is a finitely generated group having at least 2 elements. We show that for d ≥ 5, for any measure on C such that its third moment is finite and the support of the measure generates C as a group, the Poisson boundary can be identified with the limit “lamplighter” configurations on A. This provides a partial answer to a question of Kaimanovich and Vershik [44]. Also, for free metabelian groups Sd,2 on d generators, d ≥ 5, we answer a question of Vershik [56] and give a complete description of the Poisson–Furstenberg boundary for any non-degenerate random walk on Sd,2 having finite third moment. Finally, we give various examples of slowly decaying measures on wreath products with non-standard boundaries.