Abstract
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 $\mu$ of finite entropy $H(\mu)$. This implies that the entropy $h(\mu)$ 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 $\alpha\lt 1$. We show that in some of our examples the growth function satisfies $\exp(\frac{n}{\ln^{2+\varepsilon}(n)}) \le v_{G,S}(n) \le \exp(\frac{n}{\ln^{1-\varepsilon}(n)})$ for any $\varepsilon >0$ and any sufficiently large $n$.
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