×

Discrete groups and non-Riemannian homogeneous spaces. (English) Zbl 0801.22009

Let \(H\) be a Lie group, \(J \subset H\) a Lie subgroup. A discrete subgroup \(\Gamma \subset H\) is called a cocompact lattice on \(J \setminus H\) if \(\Gamma\) acts freely and properly discontinuously on \(J\setminus H\), with a compact quotient \(M\). A cocompact lattice always exists if \(J\) is compact, the corresponding manifolds \(M\) being the locally homogeneous Riemannian manifolds. The author formulates a very general condition on the pair \((H,J)\), under which no cocompact lattice exists unless \(J\) is compact. In particular, this is true in the case when \(H\) is an almost simple real algebraic group and \(J\) is a unimodular algebraic subgroup such that \(Z_ H(J)\) contains a simple Lie group of \(\mathbb{R}\)-rank at least 2 that does not have a local embedding into \(J\). As an example, \(\text{SL}(p,\mathbb{R}) \setminus \text{SL}(n,\mathbb{R})\) does not admit a cocompact lattice if \(2 \leq p < n/2\), \(n > 4\).

MSC:

22E40 Discrete subgroups of Lie groups
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
43A85 Harmonic analysis on homogeneous spaces
53C30 Differential geometry of homogeneous manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yves Benoist and François Labourie, Sur les espaces homogènes modèles de variétés compactes, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 99 – 109 (French). · Zbl 0786.53031
[2] Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111 – 122. · Zbl 0116.38603 · doi:10.1016/0040-9383(63)90026-0
[3] E. Calabi and L. Markus, Relativistic space forms, Ann. of Math. (2) 75 (1962), 63 – 76. · Zbl 0101.21804 · doi:10.2307/1970419
[4] Kevin Corlette and Robert J. Zimmer, Superrigidity for cocycles and hyperbolic geometry, Internat. J. Math. 5 (1994), no. 3, 273 – 290. · Zbl 0812.58019 · doi:10.1142/S0129167X94000176
[5] W. Goldman, Projective geometry on manifolds, Univ. of Maryland Lecture Notes, 1988.
[6] Toshiyuki Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249 – 263. · Zbl 0662.22008 · doi:10.1007/BF01443517
[7] Toshiyuki Kobayashi and Kaoru Ono, Note on Hirzebruch’s proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 71 – 87. · Zbl 0726.57019
[8] Ravi S. Kulkarni, Proper actions and pseudo-Riemannian space forms, Adv. in Math. 40 (1981), no. 1, 10 – 51. · Zbl 0462.53041 · doi:10.1016/0001-8708(81)90031-1
[9] Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545 – 607. · Zbl 0763.28012 · doi:10.2307/2944357
[10] Maxwell Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci. 35 (1963), 487 – 489. · Zbl 0123.13804
[11] Joseph A. Wolf, The Clifford-Klein space forms of indefinite metric, Ann. of Math. (2) 75 (1962), 77 – 80. · Zbl 0101.37503 · doi:10.2307/1970420
[12] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. · Zbl 0571.58015
[13] Robert J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980), no. 3, 511 – 529. · Zbl 0468.22011 · doi:10.2307/1971090
[14] Robert J. Zimmer, On the algebraic hull of an automorphism group of a principal bundle, Comment. Math. Helv. 65 (1990), no. 3, 375 – 387. · Zbl 0733.54028 · doi:10.1007/BF02566614
[15] Robert J. Zimmer, Ergodic theory and the automorphism group of a \?-structure, Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984) Math. Sci. Res. Inst. Publ., vol. 6, Springer, New York, 1987, pp. 247 – 278. · doi:10.1007/978-1-4612-4722-7_10
[16] Robert J. Zimmer, Superrigidity, Ratner’s theorem, and fundamental groups, Israel J. Math. 74 (1991), no. 2-3, 199 – 207. · Zbl 0767.22002 · doi:10.1007/BF02775786
[17] Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373 – 409. · Zbl 0334.28015
[18] R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), no. 3, 425 – 436. · Zbl 0576.22014 · doi:10.1007/BF01388637
[19] Robert J. Zimmer, Groups generating transversals to semisimple Lie group actions, Israel J. Math. 73 (1991), no. 2, 151 – 159. · Zbl 0756.28013 · doi:10.1007/BF02772946
[20] -, Topological superrigidity, preprint. · Zbl 0915.58072
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.