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

TL;DR

This paper extends the Going-Down principle from locally compact groups to ample Hausdorff groupoids using KK-theory, and applies it to relate the Baum-Connes conjecture for groupoids and their fibers.

Contribution

It generalizes the Going-Down principle to ample groupoids and applies it to the Baum-Connes conjecture, linking properties of groupoids to their fibers.

Findings

Extended Going-Down principle to ample Hausdorff groupoids.

Proved Baum-Connes conjecture for strongly amenable at infinity groupoids.

Connected the conjecture for groupoid bundles to their fibers.

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.