Homology, Homotopy and Applications

Volume 19 (2017)

Number 2

The Mayer–Vietoris sequence for graphs of groups, property (T), and the first $\ell^2$-Betti number

Pages: 251 – 274

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a13

Authors

Talia Fernós (Department of Mathematics and Statistics, University of North Carolina, Greensboro, N.C., U.S.A.)

Alain Valette (Institut de Mathématiques, Université de Neuchâtel, Switzerland)

Abstract

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.

Keywords

Mayer–Vietoris sequence, 1-cohomology, graph of groups, property (T), $\ell^2$-Betti number

2010 Mathematics Subject Classification

20E08, 20J06, 22D10

Received 26 September 2016

Received revised 2 February 2017

Published 15 November 2017