Groups with Spanier-Whitehead duality

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*-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.

28 pages (final version), to appear in Annals of K-Theory

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Cited by in corpus (7)

• Categorical approach to the Baum-Connes conjecture for étale groupoids
• On finitely summable Fredholm modules from Smale spaces
• Hofstadter-Toda spectral duality and quantum groups
• Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
• Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
• A geometric representative for the fundamental class in KK-duality of Smale spaces
• A geometric Elliott invariant and noncommutative rigidity of mapping tori