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Pseudogroups of isometries of ℝ: reconstruction of free actions on ℝ-trees

Published online by Cambridge University Press:  19 September 2008

Damien Gaboriau
Affiliation:
Unité de Mathématiques CNRS UMR 128, Ecole Normale Supérieure de Lyon, 46 allée d'ltalie, 69364 Lyon Cedex 07, France
Gilbert Levitt
Affiliation:
Laboratoire de Topologie et Géométrie CNRS URA 1408, Université Toulouse III, 118 route de Narbonne, 31062 Toulouse Cedex, France
Frédéric Paulin
Affiliation:
Unité de Mathématiques CNRS UMR 128, Ecole Normale Supérieure de Lyon, 46 allée d'ltalie, 69364 Lyon Cedex 07, France

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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