The intersection of the subgroups of finite index in Baumslag-Solitar groups.
type: Item
instance of: scholarly article
title: The intersection of the subgroups of finite index in Baumslag-Solitar groups.
zbMATH Open document ID: 1204.20041
DOI: 10.1134/S0001434610010116
author: D. I. Moldavanskiĭ
published in: Mathematical Notes
publication date: 2010-09-24T00:00:00Z
review text: For a group \(G\) let \(\sigma(G)\) denote the intersection of all finite-index normal subgroups of \(G\). Then \(\sigma(G)=1\) if and only if \(G\) is residually finite. The article first states results concerning \(\sigma(G)\) for certain classes of groups that follow from earlier papers. A result for \(\sigma(G)\) for HNN-extensions follows from: [\textit{D. I. Moldavanskii}, Ukr. Math. J. 44, No. 6, 758-760 (1992); translation from Ukr. Mat. Zh. 44, No. 6, 842-845 (1992; Zbl 0788.20014)]. Theorem. Let \(G\) be a group, \(\varphi\) an injective endomorphism of \(G\), and let \(G(\varphi)=\langle G,t\mid t^{-1}gt=g\varphi,\;(g\in G)\rangle\) be the HNN-extension of \(G\) relative to \(\varphi\). Then \(\sigma(G(\varphi))\) coincides with the normal closure in \(G(\varphi)\) of the intersection of all \(\varphi\)-compatible finite-index subgroups of \(G\). (A subgroup \(H\) of \(G\) is called \(\varphi\)-compatible if \(H\varphi=G\varphi\cap H\).) A result for \(\sigma(G)\) for generalised (amalgamated) free products follows from [\textit{D. I. Moldavanskii}, ``The intersection of the subgroups of finite-index in some generalized free products of groups.'' Vestn. Ivanovsk. Gos. Un-ta. Ser. Estestv., Obshchestv. Nauki, No. 2, 114-122 (2008)]: Theorem. Let \(A,B\) be residually finite groups and let \(H\) be a proper subgroup of \(A\), \(K\) a proper central subgroup of \(B\), \(\varphi\colon H\to K\) an isomorphism, and let \(G\) be the generalised (amalgamated) free product \(\langle A*B\mid H=K,\;\varphi\rangle\). Then \(\sigma(G)\) coincides with the normal closure in \(G\) of the mutual commutator set \([H_1,B]\), where \(H_1\) is the intersection of all finite-index subgroups of \(A\) containing the subgroup \(H\). The present paper provides a similar result for \(\sigma(G)\) for the Baumslag-Solitar groups: Theorem 1. Let \(G(m,n)\) be the Baumslag-Solitar group \(\langle a,b\mid a^{-1}b^ma=b^n\rangle\) where \(m,n\) are nonzero integers. Then \(\sigma(G(m,n))…
reviewed by: Gerald Williams
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 5791023
zbMATH Keywords: Baumslag-Solitar groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: one-relator groups
zbMATH Keywords: residual finiteness
zbMATH Keywords: normal closures
zbMATH Keywords: amalgamated products
zbMATH Keywords: HNN-extensions
MaRDI profile type: Publication
cites work: Residual finiteness of descending HNN-extensions of groups
cites work: Some two-generator one-relator non-Hopfian groups
cites work: Nonresidually Finite One-Relator Groups
cites work: The Intersection of the Subgroups of Finite Index in Some Finitely Presented Groups
full work available at URL: s0001434610010116
OpenAlex ID: W2091048392
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description: scientific article; zbMATH DE number 5791023