Skip to main content
SpringerLink
Log in

A cohomological obstruction to the existence of compact Clifford–Klein forms

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In this paper, we continue the study of the existence problem of compact Clifford–Klein forms from a cohomological point of view, which was initiated by Kobayashi–Ono and extended by Benoist–Labourie and the author. We give an obstruction to the existence of compact Clifford–Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. \({{\text {SO}}}_0(p+r, q)/({{\text {SO}}}_0(p,q) \times {{\text {SO}}}(r))\,(p,q,r \ge 1, \ q{:}\,\text {odd})\) and \({{\text {SL}}}(p+q, {\mathbb {C}})/{{\text {SU}}}(p,q)\,(p,q \ge 1)\), of a homogeneous space that does not admit a compact Clifford–Klein form. To construct these examples, we apply Cartan’s theorem on relative Lie algebra cohomology of reductive pairs and the theory of \(\varepsilon \)-families of semisimple symmetric pairs.

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. Benoist, Y.: Actions propres sur les espaces homogènes réductifs. Ann. Math. 144, 315–347 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benoist, Y., Labourie, F.: Sur les espaces homogènes modèles de variétés compactes. Inst. Hautes Études Sci. Publ. Math. 76, 99–109 (1992)

    Article  MATH  Google Scholar 

  3. Borel, A.: Compact Clifford–Klein forms of symmetric spaces. Topology 2, 111–122 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borel, A.: Semisimple Groups and Riemannian Symmetric Spaces. Texts and Readings in Mathematics, vol. 16. Hindustan Book Agency, New Delhi (1998)

    Google Scholar 

  5. Borel, A., Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math. 75, 485–535 (1962)

  6. Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 7–9. Translated from the 1975 and 1982 French originals by Andrew Pressley. Elements of Mathematics (Berlin). Springer, Berlin (2005)

  7. Calabi, E., Markus, L.: Relativistic space forms. Ann. Math. 75, 63–76 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal. In: Colloque de topologie (espaces fibrés), Bruxelles, 1950, pp. 57–71, Georges Thone, Liège; Masson et Cie., Paris (1951)

  9. Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton, NJ (1956)

    MATH  Google Scholar 

  10. Dold, A.: Partitions of unity in the theory of fibrations. Ann. Math. 78, 223–255 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  11. Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology. Volume III: Cohomology of Principal Bundles and Homogeneous Spaces. Pure and Applied Mathematics, vol. 47-III. Academic Press, New York (1976)

  12. Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. With an appendix containing two reprints by Henri Cartan. Mathematics Past and Present. Springer, Berlin (1999)

  13. Hochschild, G.: The Structure of Lie Groups. Holden-Day, Inc., San Francisco (1965)

    MATH  Google Scholar 

  14. Knapp, A.W.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser Boston, Inc., Boston, MA (2002)

    MATH  Google Scholar 

  15. Kobayashi, T.: Proper action on a homogeneous space of reductive type. Math. Ann. 285, 249–263 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kobayashi, T.: Discontinuous groups acting on homogeneous spaces of reductive type. In: Representation Theory of Lie Groups and Lie Algebras (Fuji-Kawaguchiko, 1990), pp. 59–75. World Sci. Publ., River Edge, NJ (1992)

  17. Kobayashi, T.: A necessary condition for the existence of compact Clifford–Klein forms of homogeneous spaces of reductive type. Duke Math. J. 67, 653–664 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kobayashi, T.: Discontinuous groups and Clifford–Klein forms of pseudo-Riemannian homogeneous manifolds. In: Algebraic and Analytic Methods in Representation Theory (Sønderborg, 1994), pp. 99–165, Perspect. Math., 17, Academic Press, San Diego, CA (1997)

  19. Kobayashi, T., Ono, K.: Note on Hirzebruch’s proportionality principle. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 71–87 (1990)

    MathSciNet  MATH  Google Scholar 

  20. Kobayashi, T., Yoshino, T.: Compact Clifford–Klein forms of symmetric spaces—revisited. Pure Appl. Math. Q. 1, 591–663 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kulkarni, R.S.: Proper actions and pseudo-Riemannian space forms. Adv. Math. 40, 10–51 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Labourie, F., Mozes, S., Zimmer, R.J.: On manifolds locally modelled on non-Riemannian homogeneous spaces. Geom. Funct. Anal. 5, 955–965 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Labourie, F., Zimmer, R.J.: On the non-existence of cocompact lattices for \(SL(n)/SL(m)\). Math. Res. Lett. 2, 75–77 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Margulis, G.: Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients. Bull. Soc. Math. Fr. 125, 447–456 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morita, Y.: A topological necessary condition for the existence of compact Clifford–Klein forms. J. Differ. Geom. 100, 533–545 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Morita, Y.: Homogeneous Spaces of Nonreductive Type Locally Modelling No Compact Manifold. arXiv:1508.04862v1, preprint (2015)

  27. Mostow, G.D., Tamagawa, T.: On the compactness of arithmetically defined homogeneous spaces. Ann. Math. 76, 446–463 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oh, H.: Tempered subgroups and representations with minimal decay of matrix coefficients. Bull. Soc. Math. Fr. 126, 355–380 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Okuda, T.: Classification of semisimple symmetric spaces with \(SL(2, {\mathbb{R}})\)-actions. J. Differ. Geom. 94, 301–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)

    MATH  Google Scholar 

  31. Oshima, T., Sekiguchi, J.: Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math. 57, 1–81 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Oshima, T., Sekiguchi, J.: The restricted root system of a semisimple symmetric pair. In: Group Representations and Systems of Differential Equations (Tokyo, 1982), pp. 433–497, Adv. Stud. Pure Math., 4, North-Holland, Amsterdam (1984)

  33. Rossmann, W.: The structure of semisimple symmetric spaces. Can. J. Math. 31, 157–180 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Selberg, A.: On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to Function Theory (internat. Colloq. Function Theory, Bombay, 1960), pp. 147–164. Tata Institute of Fundamental Research, Bombay (1960)

  35. Shalom, Y.: Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Ann. Math. 152, 113–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shimeno, N.: A compact imbedding of semisimple symmetric spaces. J. Math. Sci. Univ. Tokyo 3, 551–569 (1996)

    MathSciNet  MATH  Google Scholar 

  37. Tholozan, N.: Volume and Non-existence of Compact Clifford–Klein Forms. arXiv:1511.09448v1, preprint (2015)

  38. Wolf, J.A.: The Clifford–Klein space forms of indefinite metric. Ann. Math. 75, 77–80 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wolf, J.A.: Isotropic manifolds of indefinite metric. Comment. Math. Helv. 39, 21–64 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zimmer, R.J.: Discrete groups and non-Riemannian homogeneous spaces. J. Am. Math. Soc. 7, 159–168 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Yosuke Morita

Authors
  1. Yosuke Morita
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yosuke Morita.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Morita, Y. A cohomological obstruction to the existence of compact Clifford–Klein forms. Sel. Math. New Ser. 23, 1931–1953 (2017). https://doi.org/10.1007/s00029-016-0295-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-016-0295-1

Keywords

Mathematics Subject Classification

Search

Navigation