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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 ,
Gilbert Levitt  and
Frédéric Paulin
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
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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
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 4 , August 1995 , pp. 633 - 652
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[AL]Arnoux, P. and Levitt, G.. Sur l'unique ergodicité des 1-formes fermées singulières. Inv. Math. 84 (1986), 141–156.CrossRefGoogle Scholar
[AY]Arnoux, P. and Yoccoz, J.-C.. Construction de difféomorphismes pseudo-Anosov. C. R. Acad. Sci. Paris 292 (1981), 75–78.Google Scholar
[BF]Bestvina, M. and Feighn, M.. Stable actions of groups on real trees. Preprint. (February 1992).Google Scholar
[Ehr]Ehresmann, C.. Sur la théorie des espaces fibrés. Coll. Int. Top. Alg. Paris, CNRS (1947), 3–15.Google Scholar
[FLP]Fathi, A., Laudenbach, F. and Poenaru, V.. Travaux de Thurston sur les surfaces. Aslérisque. 66–67 (1979).Google Scholar
[Gab]Gaboriau, D.. Dynamique des systèmes d'isométries et actions de groupes sur les arbres réels. Thèse. Toulouse, June 1993.Google Scholar
[GL]Gaboriau, D. and Levitt, G.. The rank of actions on ℝ-trees. Ann. Scien. E.N.S. Paris. To appear.Google Scholar
[GLP1]Gaboriau, D., Levitt, G. and Paulin, F.. Pseudogroups of isometries of ℝ: Rips' theorem on free actions on ℝ-trees. Isr. J. Math. 87 (1994), 403–428.CrossRefGoogle Scholar
[Gro]Gromov, M.. Groups of polynomial growth and expanding maps. Pub. I.H.E.S. 53 (1981), 53–78.Google Scholar
[GS]Gillet, H. and Shalen, P.. Dendrology of groups in low ℚ-ranks. J. Diff. Geom. 32 (1990), 605–712.Google Scholar
[Gus]Gusmao, P.. Groupes et feuilletages de codimension 1. Thèse. Toulouse, June 1993.Google Scholar
[Hae1]Haefliger, A.. Groupoïdes d'holonomie et classifiants. Structures transverses des feuilletages. Astérisque. 116 (1984), 70–97.Google Scholar
[Hae2]Haefliger, A.. Pseudogroups of local isometries. Proc. Vth Coll. in Differential Geometry. Cordero, L. A., ed. Research Notes in Mathematics 131. Pitman: London, 1985, pp 174–197.Google Scholar
[Lev1]Levitt, G.. Groupe fondamental de l'espace des feuilles dans les feuilletages sans holonomie. J. Diff. Geom. 31 (1990), 711–761.Google Scholar
[Lev2]Levitt, G.. La dynamique des pseudogroupes de rotations. Invent. Math. 113 (1993), 633–670.Google Scholar
[Lev3]Levitt, G.. Constructing free actions on ℝ-trees. Duke Math. J. 69 (1993), 615–633.CrossRefGoogle Scholar
[LP]Levitt, G. and Paulin, F.. Geometric group actions on trees. Preprint Univ. Toulouse (Sept. 1994).Google Scholar
[Mol]Molino, P.. Riemannian Foliations. Progress in Mathematics Vol. 73. Birkhauser: Basel, 1988.Google Scholar
[Mor1]Morgan, J.. Δ-trees and their applications. Bull. Amer. Math. Soc. 26 (1992), 87–112.CrossRefGoogle Scholar
[Mor2]Morgan, J.. Notes on Rips' lectures at Columbia University, October 1991. Manuscript.Google Scholar
[MO]Morgan, J. and Otal, J.-P.. Relative growth rate of closed geodesies on a surface under varying hyperbolic structures. Comm. Math. Helv. 68 (1993), 171–208.Google Scholar
[MS1]Morgan, J. and Shalen, P.. Valuations, trees and degeneration of hyperbolic structures II, III. Ann. Math. 122 (1988), 403–519.CrossRefGoogle Scholar
[Pau1]Paulin, F.. The Gromov topology on ℝ-trees. Topology and its Appl. 32 (1989), 197–221.CrossRefGoogle Scholar
[Pau2]Paulin, F.. Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94 (1988), 53–80.Google Scholar
[Pau3]Paulin, F.. Dégénérescences algébriques de représentations hyperboliques. Proc. Coll. sur les variétés de dimension 3 (Luminy 1989). To appear.Google Scholar
[Pau4]Paulin, F.. A dynamical system approach to free actions on ℝ-trees: a survey with complements. Proc. Haifa 1992 Conf. on Geometric Topology. Contemp. Math. Amer. Math. Soc. 164 (1994), 187–217.Google Scholar
[Rim1]Rimlinger, F.. Free actions on ℝ-trees. Trans. Amer. Math. Soc. 332 (1992), 315–331.Google Scholar
[Rim2]Rimlinger, F.. ℝ-trees and normalization of pseudogroups. Exp. Math. 1 (1992), 95–114.Google Scholar
[Rim3]Rimlinger, F.. Two-complexes with similar foliations. Preprint. (1992).Google Scholar
[RS]Rourke, C. and Sanderson, M.. PL Topology. Springer: Berlin, 1985.Google Scholar
[Sac]Sacksteder, R.. Foliations and pseudogroups. Amer. J. Math. 87 (1965), 79–102.CrossRefGoogle Scholar
[Sal]Salem, E.. Riemannian foliations and pseudogroups of isometries. (Appendix in ‘Riemannian foliations’, Molino, P., Progress in Mathematics Vol. 73, Birkhäuser: Basel, 1988, pp 265–296.Google Scholar
[Shal]Shalen, P.. Dendrology of Groups: An Introduction. Essays in Group Theory. Gersten, S. M., ed. M.S.R.I. 8, Springer: Berlin, 1987.Google Scholar
[Sha2]Shalen, P.. Dendrology and its Applications. Group Theory from a Geometrical Viewpoint. Ghys, E., Haefliger, A., Verjovsky, A., eds. World Scientific: Singapore, 1991.Google Scholar
[SS]Singer, I.M. and Sternberg, S.. The infinite groups of Lie and Cartan. J. Anal. Math. 15 (1965), 1–114.Google Scholar
[VW]Veblen, O. and Whitehead, J.H.C.. The foundations of differential geometry. Camb. Tracts Math. 29 (1932).Google Scholar