Abstract
We define the Haagerup property in the general context of countable groupoids equipped with a quasi-invariant measure. One of our objectives is to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. Our second goal, concerning the general situation, is to provide a definition of this property in purely geometric terms, whereas this notion had been introduced by Ueda in terms of the associated inclusion of von Neumann algebras. Our equivalent definition makes obvious the fact that treeability implies the Haagerup property for such groupoids and that it is not compatible with Kazhdan’s property (T).
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