The \(\Sigma^2\)-conjecture for metabelian groups: the general case.
type: Item
instance of: scholarly article
title: The \(\Sigma^2\)-conjecture for metabelian groups: the general case.
zbMATH Open document ID: 1047.20025
DOI: 10.1016/S0021-8693(03)00267-9
author: Jens Harlander
author: Dessislava H. Kochloukova
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: Let \(m\) be a positive integer. Let \(G\) be a group and \(K(G,1)\) a complex with finite \(m\)-skeleton. A character \(\chi\colon G\to\mathbb{R}\) gives rise to a height function \(h\colon\overline X\to\mathbb{R}\) on the universal covering of \(X\). The geometric invariant \(\Sigma^m(G)\), consisting of the set of classes of characters under some special equivalence using these height functions, was introduced by Bieri and Strebel (1980) on finitely generated metabelian groups. There is a connection with homotopy groups of high dimensions. It was shown by them that \(\Sigma^1\) contains the information as to whether \(G\) is finitely presented. The \(\Sigma^m\)-conjecture says that for finitely generated metabelian \(FP_m\)-groups \(\Sigma^m(G)\) can be obtained from \(\Sigma^1(G)\) by a simple process. -- The authors prove the \(\Sigma^2\)-conjecture.
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 2055677
zbMATH Keywords: metabelian groups
zbMATH Keywords: \(FP_2\)-groups
zbMATH Keywords: finitely presented groups
zbMATH Keywords: Bieri-Neumann-Strebel invariants
zbMATH Keywords: characters
zbMATH Keywords: spheres
reviewed by: Vitaliĭ Roman'kov
Wikidata QID: Q122899241
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2055677