  The main goal of this paper is to determine the Poisson boundary of
lamplighter random walks over a general class of discrete groups $\Gamma$
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. A geometrical method for constructing the strip as a subset of
the lamplighter group starting with a smaller strip in the base group $\Gamma$
is developed. Then, this method is applied to several classes of base groups
$\Gamma$: 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.
