MONASTIR SEMINAR ON PROPER ACTIONS,

DISCONTINUOUS GROUPS, AND UNIFORM LATTICES

FOR PSEUDO-RIEMANNIAN HOMOGENEOUS MANIFOLDS

Toshiyuki KOBAYASHI

Department of Mathematical Sciences, University of Tokyo, Meguro, Komaba, 153, Tokyo, Japan

This note gives a bibliographic guidance of the developments in the last decade on “Clifford-Klein forms” of pseudo-Riemannian homogeneous manifolds.

Let G be a real reductive linear Lie group and H its closed subgroup. A Clifford–Klein form of a homogeneous manifold G/H is the double coset space Γ\G/H if Γ is a subgroup of G acting properly discontinuously and freely on G/H. In this case, we say Γ is a discontinuous group acting on G/H; furthermore, if Γ\G/H is compact, we say Γ is a uniform lattice for G/H. For example, any closed Riemann surface with genus ≥ 2 can be regarded as a compact Clifford– Klein form of the Poincaré plane P SL(2, ℝ)/SO(2). On the other hand, there is no compact Clifford–Klein form of the hyperboloid of one sheet P SL(2, ℝ)/SO(1, 1) (Calabi–Markus phenomenon). As is observed in the second example, not all discrete subgroup of G can act properly discontinuously on a homogeneous manifold G/H if H is non-compact. Fundamental problems are:

(1) Which homogeneous manifold G/H admits an infinite discontinuous group?

(2) Which homogeneous manifold G/H admits a compact Clifford–Klein form? First, from the view point of differential geometry, it is a basic problem to tell how

local geometric structure affects a global nature of a manifold. In this connection our problems are restated as “what is a possible fundamental group of a manifold which is locally isomorphic to a particular homogeneous manifold?” Second, from the view point of representation theory, one might expect that the affirmative case of (2) opens a theory of harmonic analysis over the Clifford-Klein forms.

(2) is always the case if H is compact, by classical results of Borel, Harish-Chandra, Mostow and Tamagawa ([3], [4], [19]).

If H is non-compact and reductive, a negative example for (1) was first obtained by Calabi–Markus [5] when G/H = SO(n, 1)/SO(n − 1, 1), and followed by Wolf ([22], [23], [24]). The reductive case was settled generally in [7], giving a criterion of the Calabi-Markus phenomenon by the rank condition ℝ-rank G = ℝ-rank H. The main idea is to establish a criterion of proper actions in the sense of Palais [20], in the reductive setting ([7], Theorem4.1). This criterion is generalized in [1], [11]. On the other hand, the solvable case was studied in [8], [10], [17].

The affirmative case of (2) such as SO(2n, 2)/SO(2n, 1) and SO(2n, 2)/U(n, 1) was proved in [7] and [15]. The idea in [7] is to find criteria that the Zariski closure

Typeset by AMS-TEX

\overline{Γ} of Γ acts properly on G/H and that \overline{Γ}\G/H is compact. A list of homogeneous manifolds G/H with compact Clifford-Klein forms by this method is given in [13].

Conversely, the negative case of (2) is studied by various approaches, such as characteristic classes, cohomology of groups, symplectic geometry, ergodic actions, representation theory and so on (cf. [1], [2], [6], [7], [8], [9], [13], [14], [15], [16], [18], [25]). Typical examples in [2], [6], [16], [25] in reductive cases are also obtained as special cases of [9], Theorem 1.5, while [14] and [18] give very different examples.

An overview in this area with a number of open problems is given in [13].

References

[1] Y. Benoist, Actions propres sur les espaces homogenes reductifs, Preprint.

[2] Y. Benoist and F. Labourie, Sur les espaces homogenes modeles de varietes compactes, I.H.E.S. Publ. Math. 76 (1992), 99–109.

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

[4] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math. 75 (1962), 485–535.

[5] E. Calabi and L. Markus, Relativistic space forms, Ann. Math. 75 (1962), 63–76.

[6] K. Corlette, Harmonic maps, rigidity and Hodge theory (1994), ICM-94, invited talk.

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

[8] , Discontinuous groups acting on homogeneous spaces of reductive type, Proceedings of the International Conference on Representation Theory of Lie Groups and Lie Algebras (1990 August-September, at Fuji-Kawaguchiko) (1992), World Scientific, Singapore, 59–75.

[9] , A necessary condition for the existence of compact Clifford–Klein forms of homogeneous spaces of reductive type, Duke Math. J. 67 (1992), 653–664.

[10] , On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geometry and Physics 12 (1993), 133–144.

[11] , Criterion of proper actions on homogeneous space of reductive groups, J. Lie Theory 6 (1996), 147-163.

[12] , Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, preprint, Mittag-Leffler Institute 32 (1996).

[13] , Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, Lecture Notes in the European School on Group Theory held at Sandbjerg Gods’, August 1994, Perspectives in Mathematics 17, Academic Press, 99-165.

[14] T. Kobayashi and K. Ono, Note on Hirzebruch’s proportionality principle, J. Fac. Soc. University of Tokyo 37 (1990), 71–87.

[15] R. S. Kulkarni, Proper actions and pseudo-Riemannian space forms, Advances in Math. 40 (1981), 10–51.

[16] F. Labourie, S. Mozes and R. J. Zimmer, On manifolds locally modelled on non-Riemannian homogeneous spaces, preliminary version (1994).

[17] R. Lipsman, Proper actions and a compactness condition, J. Lie Theory 5 (1995), 25–39.

[18] G. Margulis, Decay of matrix coefficients and existence of compact quotients of homogeneous spaces, hand-written notes (1996).

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

[20] R. S. Palais, On the existence of slices for actions of noncompact Lie groups, Ann. Math. 73 (1961), 295–323.

[21] N. R. Wallach, Two problems in the theory of automorphic forms, Open Problems in Representation Theory, 1988, pp. 39–40, Proceedings at Katata, 1986.

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

[23] , Isotropic manifolds of indefinite metric, Comment. Math. Helv. 39 (1964), 21–64.

[24] , Spaces of constant curvature, 5th ed., Publish or Perish, Inc., Wilmington, 1984.

[25] R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc. 7 (1994), 159–168.