Published online by Cambridge University Press: 19 September 2008
Rips' theorem about free actions on ℝ-trees relies on a careful analysis of finite systems of partial isometries of ℝ. We associate a free action on an ℝ-tree to any finite system of isometries without reflection. Any free action may be approximated (strongly in the sense of Gillet-Shalen) by actions arising in this way. Proofs involve the use, in an essential way, of separation properties of systems of isometries. We also interpret these finite systems of isometries as generating sets of pseudogroups of partial isometries between closed intervals of ℝ.