Journal of the European Mathematical Society


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Published online first: 2020-12-07
DOI: 10.4171/JEMS/1030

Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture

Russell Lyons[1] and Yuval Peres[2]

(1) Indiana University, Bloomington, USA
(2) Kent State University, USA

We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over $\mathbb Z^d (d \geq 3)$ is the Poisson boundary. For $d \geq 5$, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

Keywords: Random walks, free metabelian group, entropy, harmonic functions

Lyons Russell, Peres Yuval: Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture. J. Eur. Math. Soc. Electronically published on December 7, 2020. doi: 10.4171/JEMS/1030 (to appear in print)