Groups with Spanier-Whitehead Duality

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Date
2019-09-17MFO Scientific Program
OWLF 2019Series
Oberwolfach Preprints;2019,23Author
Nishikawa, Shintaro
Proietti, Valerio
Metadata
Show full item recordOWP-2019-23
Abstract
We introduce the notion of Spanier-Whitehead $K$-duality for a discrete group $G$, 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.