  We study the Poisson-Furstenberg boundary of random walks on permutational
wreath products. We give a sufficient condition for a group to admit a
symmetric measure of finite first moment with non-trivial boundary, and show
that this criterion is useful to establish exponential word growth of groups.
We construct groups of exponential growth such that all finitely supported (not
necessarily symmetric, possibly degenerate) random walks on these groups have
trivial boundary. This gives a negative answer to a question of Kaimanovich and
Vershik.
