Foliation groupoids and their cyclic homology

Abstract

In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete isotropy groups, - G is Morita equivalent to an etale groupoid. Moreover, we show that among the Lie groupoids integrating a given foliation, the holonomy and the monodromy groupoids are extreme examples. The second theorem shows that the cyclic homology of convolution algebras of foliation groupoids is invariant under Morita equivalence of groupoids, and we give explicit formulas. Combined with the previous results of Brylinski, Nistor and the authors, this theorem completes the computation of cyclic homology for various foliation groupoids, like the (full) holonomy/monodromy groupoid, Lie groupoids modeling orbifolds, and crossed products by actions of Lie groups with finite stabilizers. Some parts of the proof, such as the H-unitality of convolution algebras, apply to general Lie groupoids. Since one of our motivation is a better understanding of various approaches to longitudinal index theorems for foliations, we have added a few brief comments at the end of the second section.