Homology, Homotopy and Applications
Volume 19 (2017)
The Mayer–Vietoris sequence for graphs of groups, property (T), and the first $\ell^2$-Betti number
Pages: 251 – 274
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.
Mayer–Vietoris sequence, 1-cohomology, graph of groups, property (T), $\ell^2$-Betti number
2010 Mathematics Subject Classification
20E08, 20J06, 22D10
Paper received on 26 September 2016.
Revised paper received on 2 February 2017.