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 ${\Bbb Z}^d$ ($d \ge 3$) 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.