A 2-generated just-infinite profinite group which is not positively generated.
type: Item
instance of: scholarly article
title: A 2-generated just-infinite profinite group which is not positively generated.
zbMATH Open document ID: 1074.20024
DOI: 10.1007/BF02772214
author: Andrea Lucchini
published in: Israel Journal of Mathematics
publication date: 2004-11-11T00:00:00Z
review text: Let \(G\) be a profinite group. For a positive integer \(k\), define \(P_G(k)\) as the measure in \(G^k\) of the set of those \(k\)-tuples \((g_1,\dots,g_k)\) that generate \(G\) (topologically). \(G\) is said to be positively finitely generated (PFG) if for some \(k\) one has \(P_G(k)>0\), that is, the probability that \(k\) random elements generate \(G\) is positive. Based among others on examples of \textit{W.~M.~Kantor} and \textit{A.~Lubotzky} [Geom. Dedicata 36, No. 1, 67-87 (1990; Zbl 0718.20011)], L.~Pyber has asked whether every finitely generated profinite group which is not PFG admits an infinite epimorphic image which is PFG. In the paper under review the author shows that the answer is negative, by constructing a \(2\)-generated profinite group which is not PFG, and all of whose proper epimorphic images are finite.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 2114644
zbMATH Keywords: positively generated profinite groups
zbMATH Keywords: random elements
zbMATH Keywords: finitely generated profinite groups
zbMATH Keywords: epimorphic images
reviewed by: Andrea Caranti
MaRDI profile type: Publication
cites work: The probability of generating certain profinite groups by two elements
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cites work: Q4389857
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full work available at URL: https://doi.org/10.1007/bf02772214
OpenAlex ID: W2039700504
description: scientific article; zbMATH DE number 2114644