The geometric invariants of direct products of virtually free groups
type: Item
instance of: scholarly article
title: The geometric invariants of direct products of virtually free groups
zbMATH Open document ID: 0834.20039
DOI: 10.1007/BF02564473
author: Holger Meinert
published in: Commentarii Mathematici Helvetici
publication date: 1994-07-18T00:00:00Z
full work available at URL: https://eudml.org/doc/140325
review text: Let \(G\) be a group and let \(V(G)\) be the real vector space \(\text{Hom}(G;\mathbb{R})\). For every natural integer \(m\), there is a homological (respectively homotopical) invariant of \(G\), denoted by \(\Sigma^m(G,\mathbb{Z})\) (resp. \(\Sigma^m(G))\), which is defined as a certain conical subset of \(V(G)\). These invariants have been studied and computed for certain groups by R. Bieri and B. Renz. (We note that for the definition of the invariant \(\Sigma^m(G)\), the group is required to be of type \(F_m\), that is, that an Eilenberg-MacLane complex \(K(G,1)\) of \(G\) with finite \(m\)-skeleton exists). It is an open problem as to whether the two invariants coincide in the most general situation. In this paper, the author proves that in the case where \(G\) is equal to a finite direct product of finitely generated virtually free groups, we have, for every \(m\), \(\Sigma^m(G,\mathbb{Z})=\Sigma^m(G)\), and he gives a formula for this invariant, that is to say, a formula which describes the corresponding subset of \(V(G)\).
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55P20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 590897
zbMATH Keywords: homological invariants
zbMATH Keywords: homotopical invariants
zbMATH Keywords: groups of type \(F_ m\)
zbMATH Keywords: Eilenberg-MacLane complex
zbMATH Keywords: finite direct product of finitely generated virtually free groups
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
OpenAlex ID: W2020077392
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description: scientific article; zbMATH DE number 590897