Kazhdan's Property (T) and Property (FH) for measured groupoids

Introduced 50 years ago by David Kazhdan, Kazhdan's Property (T) has quickly become an active research area in mathematics, with a lot of important results. A few years later, this property has been generalized to discrete group actions by Robert J. Zimmer. Then, more recently, Claire Anantharaman-Delaroche has generalized it to measured groupoids.

In this work, we will continue to study this property for measured groupoids. We will introduce a Property (T) via compact sets for locally compact group actions, which is a generalization of the Property (T) for discrete group actions of R. J. Zimmer. We will develop Kazhdan's Property (T) for measured groupoids, and, for transformation group groupoids, a close link between these two properties will be proved. Then we will define and study a generalization of Property (FH) for measured groupoids. Finally, we will prove a generalization to measured groupoids of the Delorme-Guichardet theorem, which states an equivalence between Property (FH) and Property (T). This has also been proved by C. Anantharaman-Delaroche, but in a different context.