TAILIEUCHUNG - Đề tài "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 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. . | Annals of Mathematics Boundary behavior for groups of subexponential growth By Anna Erschler Annals of Mathematics 160 2004 1183 1210 Boundary behavior for groups of subexponential growth By Anna Erschler 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 1 of finite entropy H 1 . This implies that the entropy h i 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 for any a 1. We show that in some of our examples the growth function satisfies exp ln2 l n VG S n exp lnid ra for any e 0 and any sufficiently large n. 1. Introduction Let G be a finitely generated group and 1 be a probability measure on G. Consider the random walk on G with transition probabilities p x y J. x 1y starting at the identity. We say that the random walk is nondegenerate if 1 generates G as a semigroup. In the sequel we assume unless otherwise specified that the random walk is nondegenerate. The space of infinite trajectories G is equipped with the measure which is the image of the infinite product measure under the following map from G to G G x1 x2 x3 . . . x1 x1x2 x1x2x3 . . 1184 ANNA ERSCHLER Definition. Exit boundary. Let Afi be the ơ-algebra of measurable subsets of the trajectory space Gfi that are determined by the coordinates yn yn i of the trajectory y. The intersection Afi nnAfi is called the exit ơ-algebra .

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