We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all k ∈ N a torsion group K k with growth function

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 µ of finite entropy H(µ). This implies that the entropy h(µ) 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 α ) for any α < 1. We show that in some of our examples the growth function satisfies exp() for any ε > 0 and any sufficiently large n.

We prove an isoperimetric inequality for wreath products of Markov chains with variable fibers. We use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs, drift of random loops, the expected value E(exp(−tR n )), where R n is the number of distinct sites, visited up to the moment n, and, more generally, E exp − z:Lz,n =0 F (L z,n , z) , (where L z,n is the number of visits of z up to the moment n and F (x, y) is some non-negative function).