Arithmeticity of holonomy groups of Lie foliations

Zimmer, Robert J.

doi:10.1090/S0894-0347-1988-0924701-4

Arithmeticity of holonomy groups of Lie foliations

Author: Robert J. Zimmer
Journal: J. Amer. Math. Soc. 1 (1988), 35-58
MSC: Primary 22E40; Secondary 22D40
DOI: https://doi.org/10.1090/S0894-0347-1988-0924701-4
MathSciNet review: 924701
Full-text PDF Free Access

References | Similar Articles | Additional Information

[1] Hyman Bass, Finitely generated subgroups of 𝐺𝐿₂, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 127–136. MR 758465, https://doi.org/10.1016/S0079-8169(08)61638-4
[2] Robert A. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, vii, 143–158 (English, with French summary). MR 558593
[3] Armand Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188. MR 123639, https://doi.org/10.2307/1970150
[4] Yves Carrière, Feuilletages riemanniens à croissance polynômiale, Comment. Math. Helv. 63 (1988), no. 1, 1–20 (French). MR 928025, https://doi.org/10.1007/BF02566750
[5] Étienne Ghys and Yves Carrière, Relations d’équivalence moyennables sur les groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 19, 677–680 (French, with English summary). MR 802650
[6] A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, https://doi.org/10.1017/s014338570000136x
[7] C. Ehresmann, Structures feuilletees, Proc. Fifth Canadian Math. Congress, Univ. Toronto Press, Toronto, 1961, pp. 109-172.
[8] Yves Carrière, Flots riemanniens, Astérisque 116 (1984), 31–52 (French). Transversal structure of foliations (Toulouse, 1982). MR 755161
[9] Edmond Fedida, Sur les feuilletages de Lie, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A999–A1001 (French). MR 285025
[10] Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, https://doi.org/10.1090/S0002-9947-1977-0578656-4
[11] Jacob Feldman, Peter Hahn, and Calvin C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. in Math. 28 (1978), no. 3, 186–230. MR 492061, https://doi.org/10.1016/0001-8708(78)90114-7
[12] Étienne Ghys, Groupes d’holonomie des feuilletages de Lie, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 2, 173–182 (French). MR 799078
[13] André Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv. 32 (1958), 248–329 (French). MR 100269, https://doi.org/10.1007/BF02564582
[14] Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, https://doi.org/10.1090/S0002-9904-1969-12235-4
[15] G. A. Margulis, Discrete groups of motions of manifolds of non-positive curvature, Amer. Math. Soc. Transl. 109 (1977), 33-45.
[16] G. A. Margulis, Finiteness of quotient groups of discrete subgroups, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 28–39 (Russian). MR 545365
[17] Pierre Molino, Géométrie globale des feuilletages riemanniens, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 1, 45–76 (French). MR 653455
[18] Calvin C. Moore, Amenable subgroups of semisimple groups and proximal flows, Israel J. Math. 34 (1979), no. 1-2, 121–138 (1980). MR 571400, https://doi.org/10.1007/BF02761829
[19] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234
[20] Arlan Ramsay, Virtual groups and group actions, Advances in Math. 6 (1971), 253–322 (1971). MR 281876, https://doi.org/10.1016/0001-8708(71)90018-1
[21] Georges Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, 1952 (French). Publ. Inst. Math. Univ. Strasbourg 11, pp. 5–89, 155–156. MR 0055692
[22] Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, https://doi.org/10.2307/1970097
[23] Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Company of India, Ltd., Delhi, 1977. Macmillan Lectures in Mathematics, Vol. 1. MR 0578731
[24] W. Thurston, The geometry and topology of -manifolds, Princeton Univ. lecture notes.
[25] Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis 27 (1978), no. 3, 350–372. MR 0473096, https://doi.org/10.1016/0022-1236(78)90013-7
[26] Robert J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980), no. 3, 511–529. MR 595205, https://doi.org/10.2307/1971090
[27] Robert J. Zimmer, Orbit equivalence and rigidity of ergodic actions of Lie groups, Ergodic Theory Dynamical Systems 1 (1981), no. 2, 237–253. MR 661822, https://doi.org/10.1017/s0143385700009251
[28] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417
[29] R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), no. 3, 425–436. MR 735334, https://doi.org/10.1007/BF01388637
[30] C. C. Moore (ed.), Group representations, ergodic theory, operator algebras, and mathematical physics, Mathematical Sciences Research Institute Publications, vol. 6, Springer-Verlag, New York, 1987. MR 880370
[31 -, Amenable actions and dense subgroups of Lie groups, J] Robert J. Zimmer, Amenable actions and dense subgroups of Lie groups, J. Funct. Anal. 72 (1987), no. 1, 58–64. MR 883503, https://doi.org/10.1016/0022-1236(87)90081-4