Homological and topological properties of locally indicable groups
type: Item
instance of: scholarly article
title: Homological and topological properties of locally indicable groups
zbMATH Open document ID: 0533.20022
DOI: 10.1007/BF01166075
author: James Howie
author: Hans Rudolf Schneebeli
published in: Manuscripta Mathematica
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/154960
review text: This is an important paper: The authors show that three classes of groups, which have each been introduced in different contexts and for different reasons, actually coincide. This simplifies the existing theory and sheds new light on work of Adams on the Whitehead problem concerning asphericity in 2-complexes. The three classes of groups are the locally indicable groups, introduced by \textit{G. Higman} in connection with the zero divisor and unit problem in group rings [Proc. Lond. Math. Soc., II. Ser. 46, 231-248 (1940; Zbl 0025.24302)], the conservative groups, introduced by \textit{J. F. Adams} in connection with the Whitehead problem [J. Lond. Math. Soc. 30, 482-488 (1955; Zbl 0064.415)], and the Ḏ-groups introduced by \textit{R. Strebel} in connection with homological methods applied to the derived series of groups [Comment. Math. Helv. 49, 302-332 (1974; Zbl 0288.20066)]. Let \(R={\mathbb{Z}}\) or \({\mathbb{F}}_ p\). A group G is said to be locally R-indicable if every non-trivial finitely generated subgroup H admits an epimorphism \(H\twoheadrightarrow R\). G is conservative over R if \(H_ 2(K;R)=0\) for every 2-complex K acted on by G with \(H_ 2(K/G;R)=0.\) And G is a Ḏ(R)-group if \(R\otimes_{RG}\phi\) is non- injective whenever \(\phi:P\to Q\) is a non-injective homomorphism between projective RG-modules. (\(R\otimes_{RG}-\) detects injectivity). The main result, then, asserts that to be local R-indicable, to be conservative over R, and to be a Ḏ(R)-group are equivalent properties. There are also more subtle results comparing the three properties for different rings R. Applications. 1) For any group G let \(r(G)\triangleleft G\) denote the smallest normal subgroup M with G/M locally indicable. If X is a 2- complex, \(G=\pi_ 1(X),\) and \(N\triangleleft G\) a normal subgroup whose covering complex \(\tilde X\) satisfies \(H_ 2(\tilde X;{\mathbb{F}}_ p)=0\) for all p, then G/r(N) is of cohomology dimension \(\leq 2\). This applies, e.g., in the situation …
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3845837
zbMATH Keywords: Whitehead problem
zbMATH Keywords: asphericity in 2-complexes
zbMATH Keywords: locally indicable groups
zbMATH Keywords: zero divisor and unit problem in group rings
zbMATH Keywords: conservative groups
zbMATH Keywords: Ḏ-groups
zbMATH Keywords: derived series
zbMATH Keywords: finitely generated subgroup
zbMATH Keywords: projective RG-modules
zbMATH Keywords: covering complex
zbMATH Keywords: cohomology dimension
zbMATH Keywords: projective perfect \({\mathbb{Z}}G\)-module
zbMATH Keywords: descending subnormal series
zbMATH Keywords: fundamental group
zbMATH Keywords: Hurewicz map
reviewed by: Robert Bieri
MaRDI profile type: Publication
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OpenAlex ID: W1980656738
description: scientific article; zbMATH DE number 3845837