Groups acting freely on R-trees

In this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case. Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G . The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case. Let the finitely presented group G act freely on an R-tree. If G has an HNN-decomposition where ( s ) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either (a) ; or (b) , where S is a closed surface of non-positive Euler characteristic . A slightly different, more precise result is also given.