The residual finiteness of ascending HNN-extensions of certain soluble groups
type: Item
instance of: scholarly article
title: The residual finiteness of ascending HNN-extensions of certain soluble groups
zbMATH Open document ID: 1031.20017
published in: Illinois Journal of Mathematics
publication date: 2003-10-16T00:00:00Z
review text: Let \(G\) be a group and \(\varphi\) an injective endomorphism of \(G\). The corresponding HNN-extension \(G_\varphi=\langle G,t\mid g^t=g^\varphi,\;g\in G\rangle\) is called an `ascending HNN-extension'. The authors consider the interesting question of when \(G_\varphi\) is residually finite. An obvious necessary condition is that \(G\) be residually finite, but this is not sufficient, even when \(G\) is a finitely generated soluble group of derived length \(3\), as the authors show. The main results of the paper assert that \(G_\varphi\) is residually finite if \(G\) is either finitely generated Abelian-by-polycyclic-by-finite or a reduced soluble-by-finite minimax group. It is left open whether \(G_\varphi\) is residually finite when \(G\) is a finitely generated residually finite, nilpotent-by-polycyclic-by-finite group.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 1993648
zbMATH Keywords: residually finite groups
zbMATH Keywords: ascending HNN-extensions
zbMATH Keywords: finitely generated soluble groups
author: Akbar H. Rhemtulla
author: M. Shirvani
reviewed by: Derek J. S. Robinson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993648