Stable finiteness properties of infinite discrete groups. Issue 4 (8th December 2017)

• Record Type: Journal Article
• Authors: Bárcenas, Noé; Degrijse, Dieter; Patchkoria, Irakli
• Is Part Of: Journal of topology. Volume 10:Issue 4(2017)
• Page Start: 1169
• Page End: 1196
• Publication Date: 2017-12-08
• Subjects: 20J05 -- 55P42 -- 55P91 (primary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12035
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: Let G be an infinite discrete group. A classifying space for proper actions of G is a proper G ‐CW complex X such that the fixed point sets X H are contractible for all finite subgroups H of G . In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper G ‐spectra and study its finiteness properties. We investigate when G admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper G ‐spectra and to classical finiteness properties of the Weyl groups of finite subgroups of G . Finally, if the group G is virtually torsion‐free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.