Rigid Geometric Structures and Representations of Fundamental Groups

  • David Fisher


Let G be a simple Lie group, ℝ-rank(G) ≥ 2, and F < G a lattice. Assume that Γ acts analytically and ergodically on a compact manifold M preserving a volume and an analytic rigid geometric structure. In [6], we establish that either the Γ-action is isometric and π1(M) is finite or π1(M) admits a “large image” linear representation. We discuss the proof of this result. We also present related results which use similar techniques to show that under slightly stronger hypotheses the Γ-action is a 0-entropy extension of a standard arithmetic example. We give one new result in which this extension can be shown to be continuous rather than measurable.


Fundamental Group Compact Manifold Lattice Action Zariski Closure Finite Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amores, A. M., Vector fields of a finite type G-structure, J. Diff. G torn. 14 (1979) 1–6.MathSciNetMATHGoogle Scholar
  2. 2.
    Benoist, Y., Orbites des structures rigides (d’après M. Gromov), Integrable systems and foliations, Progr. Math. 145, Birkhäuser, Boston, 1997.Google Scholar
  3. 3.
    Feres, R., Rigid geometric structures and actions of semisimple Lie groups, Proceedings of Strasbourg Conference, (P. Foulon, ed.), to appear.Google Scholar
  4. 4.
    Fisher, D., On the arithmetic structure of lattice actions on compact spaces, preprint.Google Scholar
  5. 5.
    Fisher, D., Whyte, K., Continuous quotients for lattice actions on compact spaces, Geom. Dedicata, to appear.Google Scholar
  6. 6.
    Fisher, D., Zimmer, R. J., Geometric lattice actions, entropy and fundamental groups, preprint.Google Scholar
  7. 7.
    Franks, J., Anosov diffeomorphism on torii, Trans. Amer. Math. Soc. 145 (1969), 117–124.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gromov, M., Rigid transformation groups, Géométrie Différentielle (D. Bernard and Y. Choquet-Bruhat, eds.) Hermann, Paris 1988.Google Scholar
  9. 9.
    Lubotzky, A., Zimmer, R. J., Arithmetic structure of fundamental groups and actions of semisimple groups, Topology, to appear.Google Scholar
  10. 10.
    Margulis, G. and Qian, N., Local rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 121–164.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ratner, M., On Raghunathan’s measure conjectures, Ann. of Math. 134 (1991), no. 3, 545–607.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Shah, N., Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie Groups and Ergodic Theory, Tata Inst. Fund. Res., Bombay, 1998, 220-271.Google Scholar
  13. 13.
    Witte, D., Measurable Quotients of Unipotent Translations on Homogeneous Spaces, Trans. Amer. Math. Soc. 354 (1994), no. 2, 577–594.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zimmer, R. J., Representations of fundamental groups of manifolds with a semisimple transformation group, J. Amer. Math. Soc. 2 (1989), no. 2, 201–213.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Zimmer, R. J., Superrigidity, Ratner’s Theorem, and fundamental groups, Israel J. Math. 74 (1991), 199–207.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Zimmer, R. J., Automorphism groups and fundamental groups of geometric manifolds, Differential Geometry: Riemannian Geometry (Los Angeles, CA 1990), Amer. Math. Soc, Providence, RI (1993), 693-710.Google Scholar
  17. 17.
    Zimmer, R. J., Entropy and arithmetic quotients for simple automorphism groups of compact manifolds, Geom. Dedicata, to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • David Fisher
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations