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A necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type. (English) Zbl 0799.53056
A compact Clifford-Klein form of a homogeneous space \(G/H\) is defined as a quotient \(\Gamma\setminus G/H\) where \(\Gamma\) is a subgroup of \(G\) acting properly discontinuously and freely on \(G/H\) so that \(\Gamma\setminus G/H\) is compact in the quotient topology. The problem stated in the title is studied here and two (rather technical) necessary conditions are proved for the existence of a compact Clifford-Klein form of a given \(G/H\) of the reductive type.
Reviewer: O.Kowalski (Praha)

MSC:
53C30 Differential geometry of homogeneous manifolds
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[1] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups , Invent. Math. 42 (1977), 1-62. · Zbl 0373.22001
[2] Y. Benoist, P. Foulon, and F. Labourie, Flots d’Anosov à distributions stable et instable différentiables , C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 351-354. · Zbl 0754.58027
[3] M. Berger, Les espaces symétriques noncompacts , Ann. Sci. École Norm. Sup. (3) 74 (1957), 85-177. · Zbl 0093.35602
[4] R. Bieri, Homological dimension of discrete groups , Mathematics Department, Queen Mary College, London, 1976. · Zbl 0357.20027
[5] A. Borel, Compact Clifford-Klein forms of symmetric spaces , Topology 2 (1963), 111-122. · Zbl 0116.38603
[6] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups , Ann. of Math. (2) 75 (1962), 485-535. JSTOR: · Zbl 0107.14804
[7] E. Calabi and L. Markus, Relativistic space forms , Ann. of Math. (2) 75 (1962), 63-76. JSTOR: · Zbl 0101.21804
[8] S. Helgason, Differential geometry, Lie groups, and symmetric spaces , Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. · Zbl 0451.53038
[9] T. Kobayashi and K. Ono, Note on Hirzebruch’s proportionality principle , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 71-87. · Zbl 0726.57019
[10] T. Kobayashi, Proper action on a homogeneous space of reductive type , Math. Ann. 285 (1989), no. 2, 249-263. · Zbl 0662.22008
[11] T. Kobayashi, Discontinuous group in a homogeneous space of reductive type , Conference on Representation Theories of Lie Groups and Lie Algebras, Seminar Reports of Unitary Representation, vol. 10, 1990, pp. 41-45.
[12] R. S. Kulkarni, Proper actions and pseudo-Riemannian space forms , Adv. in Math. 40 (1981), no. 1, 10-51. · Zbl 0462.53041
[13] T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space , Invent. Math. 57 (1980), no. 1, 1-81. · Zbl 0434.58020
[14] T. Oshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair , Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 433-497. · Zbl 0577.17004
[15] W. Rossmann, The structure of semisimple symmetric spaces , Canad. J. Math. 31 (1979), no. 1, 157-180. · Zbl 0357.53033
[16] J.-P. Serre, Cohomologie des groupes discrets , Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Princeton Univ. Press, Princeton, N.J., 1971, 77-169. Ann. of Math. Studies, No. 70. · Zbl 0235.22020
[17] Y. L. Tong and S. P. Wang, Geometric realization of discrete series for semisimple symmetric spaces , Invent. Math. 96 (1989), no. 2, 425-458. · Zbl 0685.22008
[18] N. R. Wallach, Two problems in the theory of automorphic forms , Open Problems in Representation Theory, Proceedings, Katata, Japan, 1986, pp. 39-40.
[19] G. Warner, Harmonic analysis on semi-simple Lie groups. I , Grundlehren Math. Wiss. Einzeld. Beruck Anwend., vol. 188, Springer-Verlag, New York, 1972. · Zbl 0265.22020
[20] J. A. Wolf, The Clifford-Klein space forms of indefinite metric , Ann. of Math. (2) 75 (1962), 77-80. JSTOR: · Zbl 0101.37503
[21] J. A. Wolf, Spaces of constant curvature , 5th edition ed., Publish or Perish Inc., Houston, TX, 1984. · Zbl 0556.53033
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