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On amenability of automata groups. (English) Zbl 1268.20026

Summary: We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (mother groups) which is done by analyzing the asymptotic properties of random walks on these groups.

MSC:

20E08 Groups acting on trees
43A07 Means on groups, semigroups, etc.; amenable groups
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60G50 Sums of independent random variables; random walks
20P05 Probabilistic methods in group theory
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References:

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