Algebra/Topology seminar

Speaker: Dieter Degrijse

Title: Stable finiteness properties of infinite discrete groups

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 talk we consider the stable analogue of the classifying space for proper actions in the homotopy category of proper G-spectra and study finiteness properties of such a stable classifying space for proper actions. 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 smallness 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. If G is virtually torsion-free, we show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G, thus providing a geometric interpretation of the virtual cohomological dimension of a group. We also present and example of a group that admits a stable classifying space for proper actions of strictly smaller dimension than the dimension of any classifying space for proper actions. This is joint work is N. Barcenas and Irakli Patchkoria.