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Grigorchuk group of intermediate growth. (Italian) Zbl 1142.20308
The authors give an explicit description of the 2-group of intermediate growth introduced by R. I. Grigorchuk and its most significant properties. The existence of this group allowed to give a negative answer to a question of J. Milnor in which he asked if a finitely generated group has necessarily polynomial or exponential growth, that is, if the growth function \(\gamma(n)\) has a pattern of the type \(n^d\) or type \(a^n\), with \(d\in\mathbb{N}^+\), \(a\in(1,+\infty)\). This group furnishes also the first example of a non-elementary amenable group, and thus answers a problem proposed by M. Day. It provides an additional example of a group of Burnside.

MSC:
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
43A07 Means on groups, semigroups, etc.; amenable groups
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