A Going-Down principle for ample groupoids and the Baum-Connes conjecture

Abstract

We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoids which are strongly amenable at infinity. This result in turn is then used to relate the Baum-Connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the Baum-Connes conjecture for the fibres.

arXiv commentminor corrections, final version to appear in Adv. Math., 59 pages

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Cited by in corpus (6)

• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Going-Down functors and the Künneth formula for crossed products by étale groupoids
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• The Furstenberg Boundary of a Groupoid
• $\mathrm{K}$-theory and homotopies of twists on ample groupoids
• Groupoids decomposition, propagation and K-theory