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We explore the Mayer-Vietoris sequence developed by Chiswell for thefundamental group of a graph of groups when vertex groups satisfy somevanishing assumption on the first cohomology (e.g. property (T), or vanishingof the first $\ell^2$-Betti number). We characterize the vanishing of firstreduced cohomology of unitary representations when vertex stabilizer haveproperty (T). We find necessary and sufficient conditions for the vanishing ofthe first $\ell^2$-Betti number. We also study the associated Haagerup cocycleand show that it vanishes in first reduced cohomology precisely when the actionis elementary.

The Mayer-Vietoris Sequence for Graphs of Groups, Property (T), and the First $\ell^2$-Betti Number