Groups of intermediate subgroup growth and a problem of Grothendieck. (English) Zbl 1057.20019

Pyber, László

Duke Math. J. 121, No. 1, 169-188 (2004).

Let \(f\) be a function such that for every \(\varepsilon>0\), \(n^{\log n}\leq f(n)\leq n^{\varepsilon n}\) holds if \(n\) is sufficiently large. Suppose that \(\log f(n)/\log n\) is nondecreasing. For every such \(f\) the author constructs a 4-generator group \(G\) such that \(s_n(G)\), the number of subgroups of index at most \(n\) in \(G\), grows like \(f(n)\).
The author proves that there exist continuously many nonisomorphic 4-generator residually finite groups with isomorphic profinite completions. Also it is shown that there exist continuously many finitely generated residually finite groups \(G\) such that for some subgroup \(G_0\) the continuous homomorphism \(\widehat i\colon\widehat G_0\to\widehat G\) induced by the inclusion map \(i\) is an isomorphism of profinite completions, but \(G_0\not\cong G\).

Reviewer: Alexander Ivanovich Budkin (Barnaul)

MSC:

20E07	Subgroup theorems; subgroup growth
20E18	Limits, profinite groups
20E26	Residual properties and generalizations; residually finite groups
20F05	Generators, relations, and presentations of groups

Keywords:

subgroup growth; residually finite groups; profinite completions; finitely generated groups; subgroups of finite index

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