Deformation Spaces Associated to Compact Hyperbolic Manifolds

  • Dennis Johnson
  • John J. Millson
Part of the Progress in Mathematics book series (PM, volume 67)


In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.


Tangent Cone Congruence Subgroup Double Coset Constant Scalar Curvature Deformation Space 
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]
    B.N. Apanasov, Nontriviality of Teichmuller space for Kleinian group in space, Riemann Surfaces and Related Topics, Proceedings of the 1978 Stony Brook Conference, Annals of Math. Studies No. 97, Princeton University Press (1980), 21-31.Google Scholar
  2. [2]
    M. Berger, P. Gauduchon and E. Mazet, Le Spectre d’une Variete Riemanniene, Lecture Notes in Mathematics, 194, Springer-Verlag, New York.Google Scholar
  3. [3]
    M. Berger, Quelque formules de variation pour une structure Riemanniene, Ann. Scient. Ec. Norm. Sup., 4e serie, t · 3 (1970), 285–294.Google Scholar
  4. [4]
    D. Birkes, Orbits of linear algebraic groups, Annals of Math. 93 (1971), 459–475.CrossRefGoogle Scholar
  5. [5]
    A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122.CrossRefGoogle Scholar
  6. [6]
    A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Math. Studies No. 94, Princeton University Press (1980).Google Scholar
  7. [7]
    S.S. Chen and L. Greenberg, Hyperbolic Spaces, Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers, Academic Press (1974), 49-87.Google Scholar
  8. [8]
    P. Cohen, Decision procedures for real and p-adic fields, Comm. Pure Appl. Math., 22 (1969), 131–135.CrossRefGoogle Scholar
  9. [9]
    S.P. Eilenberg and S. MacLane, Cohomology theory in abstract groups I, Annals of Math. 48 (1947), 51–78.CrossRefGoogle Scholar
  10. [10]
    J. Gasqui and H. Goldschmidt, theoremes de dualite en geometrie conforme I and II, preprints.Google Scholar
  11. [11]
    N. Koiso, On the second derivative of the total scalar curvature, Osaka Journal 16 (1979), 413–421.Google Scholar
  12. [12]
    B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. of Math 81 (1959), 973–1032.CrossRefGoogle Scholar
  13. [13]
    W.L. Lok, Deformations of locally homogeneous spaces and Kleinian groups, thesis, Columbia University (1984).Google Scholar
  14. [14]
    J. Millson, On the first Betti number of a constant negatively curved manifold, Annals of Math. 104 (1976), 235–247.CrossRefGoogle Scholar
  15. [15]
    J. Millson and M.S. Raghunathan, Geometric construction of cohomology for arithmetic groups I, Geometry and Analysis, Papers Dedicated to the Memory of V.K. Patodi, Springer (1981), 103-123.Google Scholar
  16. [16]
    J. Morgan, Group actions on trees and the compactification of the spaces of classes of SO(n,1)-representations, preprint.Google Scholar
  17. [17]
    D. Mumford and J. Fogarty, Geometric Invariant Theory, Ergenbnisse der Mathematik und ihrer Grenzgebiete 34, Springer (1982).Google Scholar
  18. [18]
    O.T. O’Meara, Introduction to Quadratic Forms, Die Grundlehren der Mathematischen Wissenschaften, 117, Springer (1963).Google Scholar
  19. [19]
    P.E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer (1978).Google Scholar
  20. [20]
    R. Palais, On the existence of slices for actions of non-compact Lie groups, Annals of Math. (2) 73 (1961), 295–323.CrossRefGoogle Scholar
  21. [21]
    M.S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer (1972).Google Scholar
  22. [22]
    M.S. Raghunathan, On the first cohomology of discrete subgroups of semi-simple Lie groups, Amer. J. Math. 87 (1965), 103–139.CrossRefGoogle Scholar
  23. [23]
    J.P. Serre, Trees, Springer (1980).Google Scholar
  24. [24]
    D. Sullivan, Discrete conformai groups and measurable dynamics, Bull of the American Math. Soc. (new series) 6 (1982), 57–73.CrossRefGoogle Scholar
  25. [25]
    W.P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University Lecture Notes.Google Scholar
  26. [26]
    V.S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Lecture Notes in Mathematics 576, Springer.Google Scholar
  27. [27]
    C. Kourouniotis, Deformations of hyperbolic structures on manifolds of several dimensions, thesis, University of London, 1984.Google Scholar
  28. [28]
    W. Goldman and J. Millson, Local rigidity of discrete groups acting on complex hyperbolic space. To appear in Inv. Math.Google Scholar
  29. [29]
    R. Schoen, Conformai deformations of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479–495.Google Scholar
  30. [30]
    R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, Birkhauser, 1984.Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Dennis Johnson
    • 1
  • John J. Millson
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations