On discontinuous groups acting on homogeneous spaces with non-compact isotropy subgroups.(English)Zbl 0815.57029

Summary: Let $$G$$ be a Lie group and $$H$$ a closed subgroup. The action of a discrete subgroup $$\Gamma$$ of $$G$$ on $$G/H$$ is not always properly discontinuous if $$H$$ is noncompact. If the action of $$\Gamma$$ is properly discontinuous, then $$\Gamma$$ is called a discontinuous group acting on $$G/H$$. If $$G/H$$ is of reductive type, it is known that there are no infinite discontinuous groups acting on $$G/H$$ (called Calabi-Markus phenomenon) iff $$\mathbb{R}$$-rank $$G = \mathbb{R}$$-rank $$H$$. For a better understanding of discontinuous groups we are thus interested in cases (i) where $$G/H$$ is nonreductive, and (ii) where $$G/H$$ is of reductive type with $$\mathbb{R}$$-rank $$G = \mathbb{R}$$-rank $$H + 1$$. In this paper we consider the Calabi-Markus phenomenon in solvable cases of type (i). We also study discontinuous groups of reductive group manifolds for case (ii) and generalize a result of Kulkarni-Raymond to higher dimensions.

MSC:

 57S30 Discontinuous groups of transformations 57S25 Groups acting on specific manifolds 53C30 Differential geometry of homogeneous manifolds 22E40 Discrete subgroups of Lie groups
Full Text:

References:

 [1] Auslander, L., The structure of compact locally affine manifolds, Topology, 3, 131-139 (1964) · Zbl 0136.43102 [2] Borel, A., Compact Clifford-Klein forms of symmetric spaces, Topology, 2, 111-122 (1963) · Zbl 0116.38603 [3] Borel, A.; Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75, 485-535 (1962) · Zbl 0107.14804 [4] Bourbaki, N., Éléments de Mathématique, Topologie Générale (1960), Hermann: Hermann Paris, Ch. 3 · Zbl 0102.27104 [5] Calabi, E.; Markus, L., Relativistic space forms, Ann. Math., 75, 63-76 (1962) · Zbl 0101.21804 [6] Chevalley, C., On the topological structure of solvable groups, Ann. Math., 42, 668-675 (1941) · JFM 67.0077.02 [7] Goldman, W. M., Nonstandard Lorentz space forms, J. Diff. Geom., 21, 301-308 (1985) · Zbl 0591.53051 [8] Goldman, W.; Kamishima, Y., The fundamental group of a compact flat space form is virtually polycyclic, J. Diff. Geom., 19, 233-240 (1984) · Zbl 0546.53039 [9] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, (Pure Appl. Math., 80 (1978), Academic Press: Academic Press New York/London) · Zbl 0177.50601 [10] Kobayashi, T., Proper action on a homogeneous space of reductive type, Math. Ann., 285, 249-263 (1989) · Zbl 0662.22008 [12] 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) · Zbl 0799.53056 [13] Kulkarni, R. S., Proper actions and pseudo-Riemannian space forms, Adv. Math., 40, 10-51 (1981) · Zbl 0462.53041 [14] Kulkarni, R. S.; Raymond, F., 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Diff. Geom., 21, 231-268 (1985) · Zbl 0563.57004 [16] Margulis, G. A., Free completely discontinuous groups of affine transformations, Sov. Math. Dokl., 28, 435-439 (1983) · Zbl 0578.57012 [17] Milnor, J., On fundamental groups of complete affinely flat manifold, Adv. Math., 25, 178-187 (1977) · Zbl 0364.55001 [18] Mostow, G. D.; Tamagawa, T., On the compactness of arithmetically defined homogeneous spaces, Ann. Math., 76, 446-463 (1962) · Zbl 0196.53201 [19] Raghunathan, M., Discrete Subgroups of Lie Groups, (Ergebnisse der Mathematik und ihre Grenzgebiete, Vol. 68 (1972), Springer: Springer Berlin) · Zbl 0254.22005 [20] Satake, I., On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA, 42, 359-363 (1956) · Zbl 0074.18103 [21] Selberg, A., On discontinuous groups in higher-dimensional symmetric spaces, Contributions to Functional Theory, 147-164 (1960), Bombay [22] Tomanov, G., The virtual solvability of the fundamental group of a generalized Lorentz space form, J. Diff. Geom., 539-547 (1990) · Zbl 0681.57027 [23] Wallach, N. R., Two problems in the theory of automorphic forms, (Oshima, T., Open Problems in Representation Theory. Open Problems in Representation Theory, Proc. Conf. at Katata (1986), Univ. of Tokyo: Univ. of Tokyo Tokyo), 39-40 [24] Warner, G., (Harmonic Analysis on Semisimple Lie Groups, Vol. 1 (1972), Springer: Springer Berlin) [25] Wolf, J. A., The Clifford-Klein space forms of indefinite metric, Ann. Math., 75, 77-80 (1962) · Zbl 0101.37503 [26] Wolf, J. A., Isotropic manifolds of indefinite metric, Comments Math. Helv., 39, 21-64 (1964) · Zbl 0125.39203 [27] Wolf, J. A., Spaces of Constant Curvature (1984), Publish or Perish · Zbl 0162.53304 [28] Zimmer, R. J., Ergodic Theory and Semisimple Groups, (Monographs in Mathematics, Vol. 81 (1984), Birkhäuser: Birkhäuser Basle) · Zbl 0576.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.