Groups of intermediate subgroup growth and a problem of Grothendieck.
type: Item
instance of: scholarly article
title: Groups of intermediate subgroup growth and a problem of Grothendieck.
zbMATH Open document ID: 1057.20019
DOI: 10.1215/S0012-7094-04-12115-3
published in: Duke Mathematical Journal
publication date: 2004-09-27T00:00:00Z
review text: 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 \(G_0\not\cong G\).
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 2103580
zbMATH Keywords: subgroup growth
zbMATH Keywords: residually finite groups
zbMATH Keywords: profinite completions
zbMATH Keywords: finitely generated groups
zbMATH Keywords: subgroups of finite index
author: Laszló Pyber
reviewed by: Alexander Ivanovich Budkin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2103580