Some groups of type \(VF\)
type: Item
instance of: scholarly article
title: Some groups of type \(VF\)
zbMATH Open document ID: 1032.20035
author: Ian J. Leary
author: Brita E. A. Nucinkis
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: A group is of type \(\mathbf F\) if it has a finite classifying space. A group is of type \(\mathbf{VF}\) if it has a finite index subgroup of type \(\mathbf F\). \textit{M. Bestvina} and \textit{N. Brady} [Invent. Math. 129, No. 3, 445-470 (1997; Zbl 0888.20021)] give a method for constructing torsion-free groups with prescribed homological finiteness properties. Using some of the methods of this paper, the authors produce, among many other, examples of groups of type \(\mathbf{VF}\) satisfying that the centralizers of some elements of finite order are not of type \(\mathbf{VF}\), groups of type \(\mathbf{VF}\) that contain infinitely many conjugacy classes of finite subgroups. Moreover, they also provide groups for which the minimal dimension of a universal proper classifying space is strictly larger than the virtual cohomological dimension of the group. These examples show that some properties are not preserved under taking finite extensions of groups. The main result may read as follows: given a finite flag complex \(L\) and a group \(Q\) of automorphisms of \(L\), they produce a group \(\widetilde H=H_L\rtimes Q\), where \(H_L\) is provided by the Bestiva-Brady machinery, and such that the homological properties of \(\widetilde H\) are determined by the homotopy type of \(L\) and the \(P\)-fixed points in \(L\) for each \(P\leq Q\).
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 55P20
zbMATH DE Number: 1965460
zbMATH Keywords: homological finiteness properties
zbMATH Keywords: virtually of type \(F\)
zbMATH Keywords: classifying spaces
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: virtual cohomological dimension
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
full work available at URL: s00222-002-0254-7
OpenAlex ID: W2090762313
DOI: 10.1007/S00222-002-0254-7
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description: scientific article; zbMATH DE number 1965460