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A short proof of the Grigorchuk-Cohen cogrowth theorem. (English) Zbl 0681.43004

Let \(G=F_ r/N\) be a factor group of the group \(F_ r\) on free generators \(x_ 1,...,x_ r\) over a normal subgroup N. Let \(| w|\) be the length of \(w\in F_ r\) with respect to \(x_ 1,...,x_ r\), \(x_ 1^{-1},...,x_ r^{-1}\) and let \(\gamma_ n=card N_ n\), where \(N_ n=\{w\in N:\) \(| w| =n\}\). This paper contains a new short proof of a theorem of J. M. Cohen and R. I. Grigorchuk that G is amenable if and only if lim sup(\(\gamma_ n)^{1/n}=2r-1\).
Reviewer: E.Płonka

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
20F05 Generators, relations, and presentations of groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20E05 Free nonabelian groups
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A05 Measures on groups and semigroups, etc.
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References:

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