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

Abstract

We study the Poisson–Furstenberg boundary of random walks on $C=A\wr B$, where $A=Z^d$ and $B$ is a finitely generated group having at least $2$ elements. We show that for $d\ge 5$, 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 $S_{d,2}$ on $d$ generators, $d\ge 5$, we answer a question of Vershik [56] and give a complete description of the Poisson–Furstenberg boundary for any non-degenerate random walk on $S_{d,2}$ having finite third moment. Finally, we give various examples of slowly decaying measures on wreath products with non-standard boundaries.