Kobayashi, Toshiyuki Proper action on a homogeneous space of reductive type. (English) Zbl 0662.22008 Math. Ann. 285, No. 2, 249-263 (1989). An action of \(L\) on a homogeneous space \(G/H\) is investigated where \(L, H\subset G\) are reductive groups. A criterion of the properness of this action is obtained in terms of the little Weyl group of \(G\). Especially \(\mathbb{R}\)-\(\operatorname{rank}G= \mathbb{R}\)-\(\operatorname{rank} H\) iff Calabi-Markus phenomenon occurs, i.e. only finite subgroups can act properly discontinuously on \(G/H\). Then by using cohomological dimension theory of a discrete group, \(L\setminus G/H\) is proved compact iff \(d(G)=d(L)+d(H)\), where \(d(G)\) denotes the dimension of Riemannian symmetric space associated with \(G\), etc. These results apply to the existence problem of lattice in \(G/H\). Six series of classical pseudo-Riemannian homogeneous spaces are found to admit non-uniform lattice as well as uniform lattice, while some necessary condition for uniform lattice is obtained when \(\operatorname{rank} G = \operatorname{rank} H\). Reviewer: Toshiyuki Kobayashi (Tokyo) Cited in 5 ReviewsCited in 69 Documents MSC: 22E40 Discrete subgroups of Lie groups 53C30 Differential geometry of homogeneous manifolds 43A85 Harmonic analysis on homogeneous spaces 53C35 Differential geometry of symmetric spaces Keywords:action; homogeneous space; reductive groups; Weyl group; cohomological dimension; Riemannian symmetric space; pseudo-Riemannian homogeneous spaces; uniform lattice PDFBibTeX XMLCite \textit{T. Kobayashi}, Math. Ann. 285, No. 2, 249--263 (1989; Zbl 0662.22008) Full Text: DOI EuDML References: [1] [Bi] Bieri, R.: Homological dimension of discrete groups. Mathematics Notes, Queen’s Mary College, 1976 [2] [Bo] Borel, A.: Compact Clifford-Klein forms of symmetric spaces. Topology2, 111-122 (1963) · Zbl 0116.38603 [3] [B-H] Borel, A., Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math.75, 485-535 (1962) · Zbl 0107.14804 [4] [C-E] Cartan, H., Eilenberg, S.: Homological algebra. Princeton: Princeton Univ. Press 1956 · Zbl 0075.24305 [5] [C-M] Calabi, E., Markus, L.: Relativistic space forms. Ann. Math.75, 63-76 (1962) · Zbl 0101.21804 [6] [FJ] Flensted-Jensen, M.: Analysis on non-Riemannian symmetric spaces. CBMS-NSF Reg. Conf. Ser. Appl. Math. 61 (1986) [7] [He] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. Pure and Appl. Math. New York, London: Academic Press 1978 · Zbl 0451.53038 [8] [K-O] Kobayashi, T., Ono, K.: Note on Hirzebruch’s proportionality principle. preprint · Zbl 0726.57019 [9] [Ku] Kulkarni, R.S.: Proper actions and pseudo-Riemannian space forms. Adv. Math.40, 10-51 (1981) · Zbl 0462.53041 [10] [M] Mostow, G.D.: Self-adjoint groups. Ann. Math.62, 44-55 (1955) · Zbl 0065.01404 [11] [Sa] Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. USA42, 359-363 (1956) · Zbl 0074.18103 [12] [Sel] Selberg, A.: On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to function theory. pp. 147-164. Bombay 1960 [13] [Ser] Serre, J.P.: Cohomologie des groupes discr?tes. In: Annals of Math. Studies, Vol. 70, pp. 77-169. Princeton: Princeton Univ. Press 1971 [14] [Wal] Wallach, N.R.: Two problems in the theory of automorphic forms. In: Open problems in representation theory. pp. 39-40 (Proceedings held at Katata, 1986) [15] [War] Warner, G.: Harmonic analysis on semisimple Lie groups 1. Berlin Heidelberg New York: Springer 1972 [16] [Wo] Wolf, J.A.: The Clifford-Klein space forms of indefinite metric. Ann. Math.75, 77-80 (1962) · Zbl 0101.37503 [17] [Y] Yosida, K.: A theorem concerning the semisimple Lie groups, Tohoku Math. J.44, 81-84 (1938) · Zbl 0018.29802 [18] Wolf, J.A.: Spaces of constant curvature, 5-th ed. Boston: Publish of Perish 1984 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.