Old and new about treeability and the Haagerup property for measured groupoids

Claire Anantharaman-Delaroche (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

This is mainly an expository text on the Haagerup property for countable groupoids equipped with a quasi-invariant measure, aiming to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. We show that our definition is equivalent to the one given by Ueda in terms of the associated inclusion of von Neumann algebras. It makes obvious the fact that treeability implies the Haagerup property for such groupoids. For the sake of completeness, we also describe, or recall, the connections with amenability and Kazhdan property (T).

Mots clés: Measured groupoids; Haagerup property; Kazhdan property; treeability

Domaines: Algèbres d'opérateurs [math.OA]; Systèmes dynamiques [math.DS]

Claire Anantharaman-Delaroche. Old and new about treeability and the Haagerup property for measured groupoids. 2011. ⟨hal-00596887⟩