A geometric invariant of discrete groups
type: Item
instance of: scholarly article
title: A geometric invariant of discrete groups
zbMATH Open document ID: 0642.57002
DOI: 10.1007/BF01389175
author: Walter D. Neumann
author: Ralph Strebel
author: Robert Bieri
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143518
review text: This paper brings together and generalizes several hitherto disjoint lines of research. The authors associate to every finitely generated group \(G\) a sphere \(S(G)\cong S^{n-1}\), where \(n\) is the \(\mathbb{Z}\)-rank of the abelianization of \(G\) and a subset \(\Sigma\) of \(S(G)\), their geometric invariant. The invariant \(\Sigma\) generalizes the Bieri-Strebel invariant defined for metabelian groups \(G\), it is closely related to the set of all finitely generated kernels of infinite cyclic quotients, studied by \textit{W. Neumann}. In case \(G\) is the fundamental group of a 3-manifold \(Y\), \(\Sigma\) is shown to equal a set studied by \textit{W. P. Thurston} [Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)] and closely related to fibrations \(Y\to S^ 1\). One important result of the paper under review is the following theorem improving a result of Bieri and Strebel: If \(G\) is finitely presented and has no nonabelian free subgroup then \[ \Sigma\cup(-\Sigma)=S(G), \] where \(-\Sigma\) is the image of \(\Sigma\) under the antipodal map. The necessary condition for finite presentability stated in this theorem has been shown to be sufficient for center-by-metabelian \(G\) [\textit{R. Bieri} and \textit{R. Strebel}, Proc. Lond. Math. Soc., III. Ser. 41, 439-464 (1980; Zbl 0448.20029)] and for solvable \(S\)-arithmetic \(G\) over a number field provided the Schur multiplier is finitely generated, an obviously necessary condition [\textit{H. Abels}, Finite presentability of \(S\)-arithmetic groups, Lond. Math. Soc. Lect. Note Ser. 121, 128-134 (1986; Zbl 0607.20027)].
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4046701
zbMATH Keywords: surface bundles over the circle. bivariant theory
zbMATH Keywords: PL-space
zbMATH Keywords: finitely generated group
zbMATH Keywords: geometric invariant
zbMATH Keywords: Bieri-Strebel invariant
zbMATH Keywords: finitely generated kernels of infinite cyclic quotients
zbMATH Keywords: fundamental group of a 3-manifold
zbMATH Keywords: finitely presented
zbMATH Keywords: metabelian
zbMATH Keywords: solvable S-arithmetic
zbMATH Keywords: Schur multiplier
reviewed by: Herbert Abels
MaRDI profile type: Publication
cites work: Trees, valuations, and the Bieri-Neumann-Strebel invariant
cites work: Q3041316
cites work: Valuations and Finitely Presented Metabelian Groups
cites work: Q3894098
cites work: A geometric invariant for nilpotent-by-Abelian-by-finite groups
cites work: On the existence of finitely generated normal subgroups with infinite cyclic quotients
cites work: Groups of piecewise linear homeomorphisms of the real line
cites work: Q3049007
cites work: On fibering certain 3-manifolds
cites work: Q3709485
OpenAlex ID: W2025687961
description: scientific article; zbMATH DE number 4046701