Geometriae Dedicata

, Volume 87, Issue 1–3, pp 181–189 | Cite as

Continuous Quotients for Lattice Actions on Compact Spaces

  • David Fisher
  • Kevin Whyte


Let Γ < SLn(\(\mathbb{Z}\)) be a subgroup of finite index, where n ≥ 5. Suppose Γ acts continuously on a manifold M, where π1(M) = \(\mathbb{Z}\)n, preserving a measure that is positive on open sets. Further assume that the induced Γ action on H1(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ : M\(\mathbb{T}\)n that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.

large group actions rigidity ergodic theory and semisimple groups 


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  1. [B] Borel, A.: Stable Real Cohomology of Arithmetic Groups II, Progr. Math. 14 (1981).Google Scholar
  2. [F] Fisher, D.: On the arithmetic structure of lattice actions on compact manifolds, preprint.Google Scholar
  3. [FZ] Fisher, D. and Zimmer, R.: Geometric lattice actions and fundamental groups, preprint.Google Scholar
  4. [KH] Katok, A. and Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge 1995.Google Scholar
  5. [KL] Katok, A. and Lewis, J.: Global rigidity for lattice actions on tori and new examples of volume preserving actions, Israel J. Math 93 (1996), 253-280.Google Scholar
  6. [LZ1] Lubotzky, A. and Zimmer, R. J.: Arithmetic structure of fundamental groups and actions of semisimple groups, Preprint.Google Scholar
  7. [MQ] Margulis, G. and Qian, N.: Local rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, to appear in Ergodic Theory Dynam. Systems.Google Scholar
  8. [W] Witte, D.: Measurable quotients of unipotent translations on homogeneous spaces, Trans. Amer. Math. Soc. 354(2) (1994), 577-594.Google Scholar
  9. [Z1] Zimmer, R. J.: Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984.Google Scholar
  10. [Z2] Zimmer, R. J.: Actions of semisimple groups and discrete subgroups, Proc. Internat. Congr. Math., Berkeley, 1986, 1247-1258.Google Scholar
  11. [Z3] Zimmer, R. J.: Lattices in semisimple groups and invariant geometric structures on compact manifolds, In: Roger Howe (ed.), Discrete Groups in Geometry and Analysis, Birkhäuser, Boston, 1987, 152-210.Google Scholar
  12. [Z4] Zimmer, R. J.: Representations of fundamental groups of manifolds with a semisimple transformation group, J. Amer. Math. Soc. 2 (1989), 201-213.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • David Fisher
    • 1
  • Kevin Whyte
    • 2
  1. 1.Department of MathematicsYale UniversityNew HavenU.S.A.
  2. 2.Department of MathematicsUniversity of ChicagoChicagoU.S.A.

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