Cohomological dimension of groups acting on ℝ-trees

Abstract

A celebrated question of Lyndon [L] as reinterpreted by Chiswell [C], Alperin and Moss [A-M], and Imrich [I], is whether one can classify the groups which act freely on ℝ-trees. It is a classical theorem that any group which acts freely, without inversions, on a simplicial tree is free. If T is a Λ-tree for Λ ⊂ ℝ a subgroup (possibly equal to ℝ itself), it is clear that not only free groups can act freely on an ℝ-tree but that free abelian groups, such as any free abelian subgroup of ℝ itself, can act freely. In [M-S3] Morgan and Shalen showed that the fundamental groups of most compact surfaces act freely on ℝ-trees. Furthermore in [M-S2] they showed that the fundamental group of a 3-manifold can act freely on an ℝ-tree only if it is a free product of fundamental groups of surfaces and of free abelian groups. This led Morgan and Shalen to conjecture that the only finitely generated groups which act freely on an ℝ-tree are free products of fundamental groups of surfaces and free abelian groups.