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

TL;DR

This paper investigates the Mayer-Vietoris sequence for graphs of groups, focusing on conditions for vanishing first cohomology, property (T), and the first -Betti number, providing new characterizations and conditions.

Contribution

It offers new criteria for the vanishing of first reduced cohomology and -Betti numbers in graphs of groups with property (T) or vanishing first -Betti number, extending existing theory.

Findings

Characterizes vanishing of first reduced cohomology for vertex groups with property (T)

Provides necessary and sufficient conditions for the vanishing of the first -Betti number

Shows the Haagerup cocycle vanishes in first reduced cohomology when the action is elementary

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 stabilizer 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.