  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*-algebras which are naturally attached to the group, namely the
reduced group C*-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.
