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Medientyp:
E-Artikel
Titel:
Boundary Behavior for Groups of Subexponential Growth
Beteiligte:
Erschler, Anna
Erschienen:
Princeton University Press, 2004
Erschienen in:
Annals of Mathematics, 160 (2004) 3, Seite 1183-1210
Sprache:
Englisch
ISSN:
0003-486X
Entstehung:
Anmerkungen:
Beschreibung:
<p> 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<tex-math>$\text{exp}(n^{\alpha})$</tex-math>for any α < 1. We show that in some of our examples the growth function satisfies<tex-math>$\text{exp}(\frac{n}{\text{ln}^{2+\varepsilon}(n)})\leq v_{G,S}(n)\leq \text{exp}(\frac{n}{\text{ln}^{1-\varepsilon}(n)})$</tex-math>for any ε > 0 and any sufficiently large n. </p>