Stable finiteness properties of infinite discrete groups

Noé Bárcenas, Dieter Degrijse, Irakli Patchkoria* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticle

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.
Original languageEnglish
Pages (from-to)1169-1196
Number of pages28
JournalJournal of Topology
Volume10
Issue number4
Early online date8 Dec 2017
DOIs
Publication statusPublished - Dec 2017

Keywords

  • 20J05
  • 55P42
  • 55P91 (primary)

Fingerprint Dive into the research topics of 'Stable finiteness properties of infinite discrete groups'. Together they form a unique fingerprint.

  • Cite this