Boundary behavior for groups of subexponential growth

In this paper we introduce a method for partial description of the Poisson boundary for a certain class of groups acting on a segment. As an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric (infinitely supported) measure p of finite entropy H(p). This implies that the entropy h(p) of the corresponding random walk is (finite and) positive. As another application we exhibit certain discontinuity for the recurrence property of random walks. Finally, as a corollary of our results we get new estimates from below for the growth function of a certain class of Grigorchuk groups. In particular, we exhibit the first example of a group generated by a finite state automaton, such that the growth function is subexponential, but grows faster than exp(n(alpha)) for any alpha < 1. We show that in some of our examples the growth function satisfies exp(n/In2+epsilon(n)) < upsilon(G,S)(n) <= exp(n/In1-epsilon(n)) for any epsilon > 0 and any sufficiently large n.