Stable finiteness properties of infinite discrete groups. Issue 4 (8th December 2017) - Bárcenas, Noé; Degrijse, Dieter; Patchkoria, Irakli
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.