Separability properties of certain tree products of groups
type: Item
instance of: scholarly article
title: Separability properties of certain tree products of groups
zbMATH Open document ID: 1006.20023
author: Goansu Kim
author: C. Y. Tang
published in: Journal of Algebra
publication date: 2002-09-08T00:00:00Z
review text: The authors study double coset separability and conjugacy separability properties of an amalgamated free product with amalgamating subgroup which is central in one of the factors. A group \(G\) said to be conjugacy separable if any two non-conjugate elements of \(G\) are non-conjugate in some finite quotient of \(G\). A subgroup \(H\) of a group \(G\) is said to be separable if for every \(g\not\in H\) there is a finite quotient of \(G\), where the image of \(H\) does not contain the image of \(g\); equivalently \(H\) is separable if \(H\) is closed in the profinite topology of \(G\). The authors call a group \(G\) central subgroup separable if every finitely generated central subgroup is separable in \(G\). Among others the following results are proved in the paper. \(\bullet\) Let \(A\), \(B\) be conjugacy separable and central subgroup separable groups and \(H\) a finitely generated common central subgroup of \(A\) and \(B\). Then the free amalgamated product \(G=A*_HB\) is conjugacy separable. \(\bullet\) Let \(G\) be a tree product of finitely many central subgroup separable and conjugacy separable groups amalgamating finitely generated central edge groups. Then \(G\) is conjugacy separable. -- As immediate consequence of this the authors deduce that a tree product of polycyclic groups amalgamating finitely generated central edge groups is conjugacy separable.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 1797475
zbMATH Keywords: conjugacy separability
zbMATH Keywords: subgroup separability
zbMATH Keywords: amalgamated free products
zbMATH Keywords: tree products
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
full work available at URL: jabr.2001.9134
OpenAlex ID: W2015696732
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DOI: 10.1006/JABR.2001.9134
description: scientific article; zbMATH DE number 1797475