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

Abstract

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 ${\mathbb{Z}}^d(d\ge 3)$ is the Poisson boundary. For $d\ge 5$, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.