A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich (Ann. Math. 152:659–692, 2000). A geometrical method for constructing the strip as a subset of the lamplighter group ${mathbb {Z}_{2}wr Gamma}$ starting with a “smaller” strip in the group Γ is developed. Then, this method is applied to several classes of base groups Γ: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.