Poisson boundaries of lamplighter groups: proof of the Kaimanovich-Vershik conjecture (Q2031664)

Summary: 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.