Commentarii Mathematici Helvetici


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Volume 86, Issue 1, 2011, pp. 113–143
DOI: 10.4171/CMH/220

Published online: 2010-12-05

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

Anna Erschler[1]

(1) École Normale Supérieure, Paris, France

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.

Keywords: Random walks on groups, Poisson boundary, Furstenberg boundary, lamplighter group, free metabelian group, solvable group, amenable group, entropy criterion, conditional entropy, exchangeable sigma-algebra

Erschler Anna: Poisson–Furstenberg boundary of random walks on wreath products and free metabelian groups. Comment. Math. Helv. 86 (2011), 113-143. doi: 10.4171/CMH/220