Abstract
Let G be an infinite discrete group. A classifying space for proper actions of G is a proper GCWcomplex 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 Gspectra 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 Gspectra and to classical finiteness properties of the Weyl groups of finite subgroups of G. Finally, if the group G is virtually torsionfree 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 language  English 

Pages (fromto)  11691196 
Number of pages  28 
Journal  Journal of Topology 
Volume  10 
Issue number  4 
Early online date  8 Dec 2017 
DOIs  
Publication status  Published  Dec 2017 
Keywords
 20J05
 55P42
 55P91 (primary)
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Irakli Patchkoria
Person: Academic