A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups.
type: Item
instance of: scholarly article
title: A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups.
zbMATH Open document ID: 1064.20028
DOI: 10.5565/PUBLMAT_47203_10
author: Thierry Coulbois
author: Mark Sapir
author: Pascal Weil
published in: Publicacions Matemàtiques
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/41465
review text: Let \(\varphi\colon F(A)\to F(B)\) be an embedding of free groups of finite ranks. Then \(\varphi\) induces a continuous epimorphism \(\widehat\varphi\colon\widehat F_{\mathbf V}(A)\to\widehat F_{\mathbf V}(B)\) of the pro-\(\mathbf V\) completions, where \(\mathbf V\) is a pseudovariety of finite groups. The paper is concerned with the injectivity of \(\widehat\varphi\). The main result is the following Theorem. Let \(\mathbf V\) be a pseudovariety of finite groups such that free groups are residually \(\mathbf V\) and let \(H=\varphi(F(A))\). Then the following conditions are equivalent: \(\bullet\) \(\widehat\varphi\) is injective. \(\bullet\) The pro-\(\mathbf V\) topology of \(H\) coincides with the topology induced by the pro-\(\mathbf V\) topology of \(F(B)\). -- If, in addition, \(\mathbf V\) is extension closed, these properties are equivalent to: \(\bullet\) \(H\) and the closure \(\text{Cl}(H)\) of \(H\) in \(F(B)\) have the same rank.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 2109795
zbMATH Keywords: profinite groups
zbMATH Keywords: pseudovarieties of finite groups
zbMATH Keywords: free groups
Wikidata QID: Q114014614
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
OpenAlex ID: W2028004159
description: scientific article; zbMATH DE number 2109795