Bárcenas, Noé; Degrijse, Dieter; Patchkoria, Irakli Stable finiteness properties of infinite discrete groups. (English) Zbl 1485.55014 J. Topol. 10, No. 4, 1169-1196 (2017). Summary: 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. Cited in 4 Documents MSC: 55P91 Equivariant homotopy theory in algebraic topology 20J05 Homological methods in group theory 55P42 Stable homotopy theory, spectra Keywords:infinite discrete group; classifying space; proper action; virtual cohomological dimension PDFBibTeX XMLCite \textit{N. Bárcenas} et al., J. Topol. 10, No. 4, 1169--1196 (2017; Zbl 1485.55014) Full Text: DOI arXiv Link