Skip to main content
SpringerLink
Log in

Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give a proof of Rips’ theorem that a finitely generated group acting freely on an ℝ-tree is a free product of free abelian groups and surface groups, using methods of dynamical systems and measured foliations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Adams,Trees and amenable equivalence relations, Ergodic Theory and Dynamical Systems10 (1990), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Arnoux and G. Levitt,Sur l’unique ergodicité des 1-formes fermées singulières, Inv. Math.84 (1986), 141–156.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Arnoux and J.-C. Yoccoz,Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sc. Paris292 (1981), 75–78.

    MATH  MathSciNet  Google Scholar 

  4. M. Bestvina and M. Feighn,Stable actions of groups on real trees, preprint (Feb. 1992).

  5. M. Culler and J. Morgan,Group actions on ℝ-trees, Proc. Lond. Math. Soc.55 (1987), 571–604.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Danthony and A. Nogueira,Measured foliations on nonorientable surfaces, Ann. Scient. Ec. Norm. Sup.23 (1990), 469–494.

    MATH  MathSciNet  Google Scholar 

  7. A. Fathi, F. Laudenbach and V. Poenaru,Travaux de Thurston sur les surfaces, Astérisque66–67, Soc. Math. France (1977).

  8. D. Gaboriau,Dynamique des systèmes d’isométries et actions de groupes sur les arbres réels, Thèse, Toulouse, June 1993.

  9. D. Gaboriau, G. Levitt and F. Paulin,Pseudogroups of isometries of ℝand constructions of ℝ-trees, Ergodic Theory & Dynamical Systems, to appear.

  10. K. W. Gruenberg,Residual properties of infinite soluble groups, Proc. London Math. Soc.7 (1957), 29–62.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Gillet and P. Shalen,Dendrology of groups in low ℚ-ranks, J. Diff. Geom.32 (1990), 605–712.

    MATH  MathSciNet  Google Scholar 

  12. P. Gusmao,Groupes et feuilletages de codimension 1, Thèse, Toulouse, June 1993.

  13. A. Haefliger,Groupoïdes d’holonomie et classifiants, inStructures transverses des feuilletages, Astérisque116, Soc. Math. France (1984), 70–97.

    MathSciNet  Google Scholar 

  14. A. Haefliger,Pseudogroups of local isometries, inProc. Vth Coll. in Differential Geometry (L.A. Cordero, ed.), Research Notes in Math131, Pitman, London, 1985, pp. 174–197.

    Google Scholar 

  15. H. Imanishi,On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ.19 (1979), 285–291.

    MATH  MathSciNet  Google Scholar 

  16. M. Keane,Interval exchange transformations, Math. Z.141 (1975), 25–31.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Levitt, 1-formes fermées singulières et groupe fondamental, Inv. Math.88 (1987), 635–667.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Levitt,Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie, J. Diff. Geom.31 (1990), 711–761.

    MATH  MathSciNet  Google Scholar 

  19. G. Levitt,La dynamique des pseudogroupes de rotations, Inv. Math.113 (1993), 633–670.

    Article  MATH  MathSciNet  Google Scholar 

  20. G. Levitt,Constructing free actions on ℝ-trees, Duke Math. J.69 (1993), 615–633.

    Article  MATH  MathSciNet  Google Scholar 

  21. G. Levitt,On the cost of generating an equivalence relation, preprint.

  22. G. S. Makanin,Equations in a free group, Math. USSR Izv.21 (1983), 145–163.

    Article  Google Scholar 

  23. J. Morgan, Λ-trees and their applications, Bull. Am. Math. Soc.26 (1992), 87–112.

    MATH  Google Scholar 

  24. J. Morgan,Notes on Rips’ lectures at Columbia University, October 1991, manuscript.

  25. J. Morgan and P. Shalen,Valuations, trees and degeneration of hyperbolic structures II, III, Ann. Math.122 (1988), 403–519.

    MathSciNet  Google Scholar 

  26. J. Morgan and P. Shalen,Free actions of surface groups on ℝ-trees, Topology30 (1991), 143–154.

    Article  MATH  MathSciNet  Google Scholar 

  27. [Pau]F. Paulin,A dynamical system approach to free actions on ℝ-trees: a survey with complements, to appear in “Proceedings of the Haifa 1992 Conference on Geometric Topology”, Contemp. Math., Amer. Math. Soc.

  28. A. A. Razborov,On systems of equations in a free group, Math. USSR Izv.25 (1985), 115–162.

    Article  Google Scholar 

  29. F. Rimlinger,Free actions on ℝ-trees, Trans. Am. Math. Soc.332 (1992), 315–331.

    Article  MathSciNet  Google Scholar 

  30. F. Rimlinger, ℝ-trees and normalization of pseudogroups, Experimental Mathematics1 (1992), 95–114.

    MATH  MathSciNet  Google Scholar 

  31. F. Rimlinger,Two-complexes with similar foliations, preprint (1992).

  32. P. Shalen,Dendrology of groups: an introduction, inEssays in Group Theory (S.M. Gersten, ed.), M.S.R.I Publ.8, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  33. P. Shalen,Dendrology and its applications, inGroup Theory from a Geometrical Viewpoint (E. Ghys, A. Haefliger and A. Verjovsky, eds.), World Scientific, Singapore, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Topologie et Géométrie C.N.R.S. URA 1408, Université Toulouse III, 118 route de Narbonne, 31062, Toulouse Cedex, France

    D. Gaboriau & G. Levitt

  2. Unité de Mathématiques C.N.R.S. UMR 128, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    F. Paulin

Authors
  1. D. Gaboriau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Levitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Paulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gaboriau, D., Levitt, G. & Paulin, F. Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees. Israel J. Math. 87, 403–428 (1994). https://doi.org/10.1007/BF02773004

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773004

Keywords

Search

Navigation