  Let T be the homogeneous tree with degree and G a finitely generated group
whose Cayley graph is T. The associated lamplighter group is the wreath product
of the cyclic group of order r with G. For a large class of random walks on
this group, we prove almost sure convergence to a natural geometric boundary.
If the probability law governing the random walk has finite first moment, then
the probability space formed by this geometric boundary together with the limit
distribution of the random walk is proved to be maximal, that is, the Poisson
boundary. We also prove that the Dirichlet problem at infinity is solvable for
continuous functions on the active part of the boundary, if the lamplighter
"operates at bounded range".
