The root class residuality of the tree product of groups with amalgamated retracts
type: Item
instance of: scholarly article
title: The root class residuality of the tree product of groups with amalgamated retracts
zbMATH Open document ID: 1450.20004
DOI: 10.1134/S0037446619040153
author: E. A. Tumanova
published in: Siberian Mathematical Journal
publication date: 2019-11-27T00:00:00Z
review text: According to \textit{K. W. Gruenberg} [Proc. Lond. Math. Soc. (3) 7, 29--62 (1957; Zbl 0077.02901)], a non trivial class \(\mathcal{K}\) of groups is called a root class whenever \(\mathcal{K}\) is closed under subgroups and direct products of finitely many factors, as well as it satisfies the following condition: If \(1 \leq Z \leq Y \leq X\) is subnormal series of \(X\) whose factors \(X/Y\) and \(Y/Z\) lie in \(\mathcal{K}\) then \(X\) has a normal subgroup \(T\) such that \(T \leq Z\) and \(X/T \in \mathcal{K}\). By an equivalent definition (see [ Comm. Algebra, vol. 43, 856-860 (2015). ]) it follows that the intersection of root classes yields a root class. It is known (see [\textit{D. N. Azarov} and \textit{E. A. Tumanova}, ``On the residuality of generalized free products by root classes of groups,'' Nauch. Tr. Ivanovsk. Gos. Univ. 6, 29--42 (2008)], with generalizations in [\textit{E. A. Tumanova}, ``On residuality of generalized free products by root classes of groups,'' Model. Anal. Inform. Sist. 20, No. 1, 133--137 (2013);\textit{E. V. Sokolov} and \textit{E. A. Tumanova}, Sib. Math. J. 57, No. 1, 135--144 (2016; Zbl 1342.20035); translation from Sib. Mat. Zh. 57, No. 1, 171--185 (2016)]) that, for every root class \(\mathcal{K}\), the free product of two residually \(\mathcal{K}\)-groups with amalgamated subgroups which are a retract in the factor groups is again a residually \(\mathcal{K}\)-group. Recall that a subgroup \(Y\) of a group \(X\) is called a retract whenever \(X\) has a normal subgroup \(Z\) with \(X=YZ\) and \(Y\cap Z=1\). Equivalently \(Y\) is a retract in \(X\) if and only if \(X\) is the split extension of some group \(Z\) by \(Y\). In the present paper, a serious generalization of the result above is obtained. Theorem A. (Theorem 2 in the paper) Consider an arbitrary root class \(\mathcal{K}\) of groups, a tree \(T=(V,E)\), and the tree product \(G\) of groups \(A_{v} \) for \(v\in V\) with subgroups \(H_{vw}\) and \(H_{wv}\) amalgamat…
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 7137342
zbMATH Keywords: tree product of groups
zbMATH Keywords: HNN-extension
zbMATH Keywords: Artin group
zbMATH Keywords: Coxeter group
zbMATH Keywords: root class residuality
zbMATH Keywords: residual finiteness
zbMATH Keywords: residual \(p\)-finiteness
zbMATH Keywords: residual solubility
MaRDI profile type: Publication
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full work available at URL: s0037446619040153
OpenAlex ID: W2968748383
description: scientific article; zbMATH DE number 7137342