R-TREES AND THE BIERI--NEUMANN-STREBEL INVARIANT

Let G be a finitely generated group. We give a new characteriza-tion of its Bieri- Neumann-Strebel invariant E(G) , in terms of geo-metric abelian actions on R-trees. We provide a proof of Brown's characterization of E(G) by exceptional abelian actions of G, using geometric methods. In a 1987 paper at Inventiones [BNS], Bieri, Neumann and Strebel associated an invariant E = E(G) to any finitely generated group G. This invariant may be viewed as a positively homogeneous open subse t of Hom(G, R) ~ {O}. It contains information about finitely generate