Duke Mathematical Journal

On Livšic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups

Edward R. Goetze and Ralf J. Spatzier

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J., Volume 88, Number 1 (1997), 1-27.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58F15
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 57S20: Noncompact Lie groups of transformations


Goetze, Edward R.; Spatzier, Ralf J. On Livšic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups. Duke Math. J. 88 (1997), no. 1, 1--27. doi:10.1215/S0012-7094-97-08801-3. https://projecteuclid.org/euclid.dmj/1077241397

Export citation


  • [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969.
  • [2] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J., 1980.
  • [3] R. Feres, Connection preserving actions of lattices in $\rm SL\sb n\bf R$, Israel J. Math. 79 (1992), no. 1, 1–21.
  • [4] H. Furstenberg, Rigidity and cocycles for ergodic actions of semisimple Lie groups (after G. A. Margulis and R. Zimmer), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin, 1981, pp. 273–292.
  • [5] E. R. Goetze, Actions of superrigid non-Kazhdan lattices on compact manifolds, Geom. Dedicata 53 (1994), no. 3, 281–285.
  • [6] E. R. Goetze, Connection preserving actions of connected and discrete Lie groups, J. Differential Geom. 40 (1994), no. 3, 595–620.
  • [7] E. R. Goetze and R. J. Spatzier, Bundle theoretic versions of Livšic's theory, in preparation.
  • [8] E. R. Goetze and R. J. Spatzier, Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices, preprint.
  • [9] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977.
  • [10] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978.
  • [11] S. Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. (2) 135 (1992), no. 2, 361–410.
  • [12] A. B. Katok and J. W. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions, Israel J. Math. 93 (1996), 253–280.
  • [13] A. B. Katok and J. W. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math. 75 (1991), no. 2-3, 203–241.
  • [14] A. B. Katok, J. W. Lewis, and R. J. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori, Topology 35 (1996), no. 1, 27–38.
  • [15] J. W. Lewis, Infinitesimal rigidity for the action of $\rm SL(n,\bf Z)$ on $\bf T\sp n$, Trans. Amer. Math. Soc. 324 (1991), no. 1, 421–445.
  • [16] A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.
  • [17] R. Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987.
  • [18] C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972), 1–23.
  • [19] N. Qian, Smooth conjugacy for Anosov diffeomorphisms and rigidity of Anosov actions of higher rank lattices, preprint.
  • [20] N. Qian, Tangential flatness and global rigidity of higher rank lattice actions, to appear in Trans. Amer. Math. Soc.
  • [21] N. Qian, Topological deformation rigidity of higher rank lattice actions, Math. Res. Lett. 1 (1994), no. 4, 485–499.
  • [22] W. Rudin, Real and Complex Analysis, 3rd ed. ed., McGraw-Hill Book Co., New York, 1987.
  • [23] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, vol. 81, Birkhäuser, Basel, 1984.
  • [24] R. J. Zimmer, On the algebraic hull of an automorphism group of a principal bundle, Comment. Math. Helv. 65 (1990), no. 3, 375–387.
  • [25] R. J. Zimmer, Topological superrigidity, preprint.