Geodesic
The Mayer–Vietoris sequence for graphs of groups, property (T), and the first $\ell^2$-Betti number
Homology, homotopy, and applications, Tome 19 (2017) no. 2, pp. 251-274.

Voir la notice de l'article provenant de la source International Press of Boston

We explore the Mayer–Vietoris sequence developed by Chiswell for the fundamental group of a graph of groups when vertex groups satisfy some vanishing assumption on the first cohomology (e.g. property (T), or vanishing of the first $\ell^2$-Betti number). We characterize the vanishing of first reduced cohomology of unitary representations when vertex stabilizers have property (T). We find necessary and sufficient conditions for the vanishing of the first $\ell^2$-Betti number. We also study the associated Haagerup cocycle and show that it vanishes in first reduced cohomology precisely when the action is elementary.
DOI : 10.4310/HHA.2017.v19.n2.a13
Classification : 20E08, 20J06, 22D10
Keywords: Mayer–Vietoris sequence, 1-cohomology, graph of groups, property (T), $\ell^2$-Betti number 
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     author = {Talia Fern\'os and Alain Valette},
     title = {The {Mayer{\textendash}Vietoris} sequence for graphs of groups, property {(T),} and the first $\ell^2${-Betti} number},
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     pages = {251--274},
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Talia Fernós; Alain Valette. The Mayer–Vietoris sequence for graphs of groups, property (T), and the first $\ell^2$-Betti number. Homology, homotopy, and applications, Tome 19 (2017) no. 2, pp. 251-274. doi : 10.4310/HHA.2017.v19.n2.a13. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2017.v19.n2.a13/

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