Topological Property (T) for groupoids
Annales de l'Institut Fourier, Online first, 52 p.

We introduce a notion of topological Property (T) for étale groupoids. This simultaneously generalizes Kazhdan’s Property (T) for groups and geometric Property (T) for coarse spaces. One main goal is to use this Property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid C * -algebras, and explore applications of this to exactness, K-exactness, and the Baum–Connes conjecture. We also study various examples, and discuss the relationship with other notions of Property (T) for groupoids and with a-T-menability.

Nous définissons une notion de propriété (T) pour les groupoïdes étales. Elle généralise à la fois la propriété (T) de Kazhdan pour les groupes, et la propriété (T) géométrique pour les espaces grossiers. Notre but principal est l’application de cette propriété (T) à l’existence de projecteurs de type Kazhdan dans les C * -algèbres réduites et maximales des groupoïdes, dont nous explorons les conséquences sur l’exactitude, l’exactitude en K-théorie, et sur la validité de la conjecture de Baum–Connes. Nous étudions aussi divers exemples, et comparons cette notion à d’autres versions de la propriété (T) ainsi qu’à la a-T-moyennabilité.

Online First:
DOI: 10.5802/aif.3513
Classification: 22A22,  46L85,  46L80,  51F99
Keywords: Property (T), Topological groupoids, Coarse geometry, Expander
Dell’Aiera, Clément 1; Willett, Rufus 2

1 ENS Lyon, UMPA Department of Mathematics 46 allée d’Italie 69342 Lyon Cedex 07 (France)
2 Department of Mathematics University of Hawai‘i at Mānoa 2565 McCarthy Mall Honolulu, HI 96822 (USA)
     author = {Dell{\textquoteright}Aiera, Cl\'ement and Willett, Rufus},
     title = {Topological {Property~(T)} for groupoids},
     journal = {Annales de l'Institut Fourier},
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     year = {2022},
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     language = {en},
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Dell’Aiera, Clément; Willett, Rufus. Topological Property (T) for groupoids. Annales de l'Institut Fourier, Online first, 52 p.

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