Abstract

Building on work by Kasparov, we study the notion of Spanier–Whitehead $K$-duality for a discrete group. It is defined as duality in the $KK$-category between two $C^{\ast }$-algebras which are naturally attached to the group, namely the reduced group $C^{\ast }$-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum–Connes conjecture by constructing duality classes based on two methods: the standard “gamma element” technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier–Whitehead duality for a large class of groups, including Bieberbach’s space groups, groups acting on trees, and lattices in Lorentz groups.

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Mathematical Subject Classification 2010

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