$\mathbf{R}$-trees and the Bieri-Neumann-Strebel invariant

Let $G$ be a finitely generated group. We give a new characterization of its Bieri-Neumann-Strebel invariant $\Sigma (G)$, in terms of geometric abelian actions on ${\bold R}$-trees. We provide a proof of Brown's characterization of $\Sigma (G)$ by exceptional abelian actions of $G$, using geometric methods.