Browsing by Author "Proietti, Valerio"

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    Groups with Spanier-Whitehead Duality
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Nishikawa, Shintaro; Proietti, Valerio
    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.
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    Homology and K-Theory of Torsion-Free Ample Groupoids and Smale Spaces
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2020) Proietti, Valerio; Yamashita, Makoto
    Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the groupoid C*-algebra when the groupoid has torsion-free stabilizers and satisfies the strong Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture by Meyer and Nest. For the unstable equivalence relation of a Smale space with totally disconnected stable sets, this spectral sequence shows Putnam's homology groups on the second sheet.