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Boundary behavior for groups of subexponential growth. (English) Zbl 1089.20025

This paper introduces a method for partial description of the Poisson boundary for a certain class of groups acting on an interval, including in particular Grigorchuk’s examples of finitely generated groups of intermediate (i.e., superpolynomial, but subexponential) growth. The main result implies that these groups admit nontrivial bounded harmonic functions with respect to a symmetric (infinitely supported) measure of finite entropy. The author also gives a certain discontinuity for the recurrence property of random walks. A further application is a nice way of estimating the rate of growth. In particular, a group generated by a finite state automaton of subexponential growth but growing faster than \(\exp(n^\alpha)\) for any \(\alpha<1\) is displayed.

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
60G50 Sums of independent random variables; random walks
43A05 Measures on groups and semigroups, etc.
60J50 Boundary theory for Markov processes
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