×

zbMATH — the first resource for mathematics

Ergodic theory, group representations, and rigidity. (English) Zbl 0532.22009
In a recent paper the author proved [Ann. Math., II. Ser. 112, 511-529 (1980; Zbl 0468.22011)] (see also H. Furstenberg [Lect. Notes Math. 842, 273-292 (1981; Zbl 0471.22007)] a remarkable result which in particular asserts that no two free ergodic finite-measure-preserving actions of two connected noncompact centerfree simple Lie groups of R- ran\(k\geq 2\) are orbit-equivalent, unless the groups are isomorphic and the actions correspond to each other under appropriate isomorphisms of the groups and the measure spaces respectively. (We recall that two actions are said to be orbit-equivalent if modulo null-sets there exists an isomorphism of the measure spaces which maps orbits of one of the actions onto orbits of the other.) A similar assertion holds for actions of lattices in these Lie groups and also in a more general set-up that we shall not go into. This kind of rigidity of ergodic actions is sharp contrast with the situation for actions of amenable groups; indeed any two free properly ergodic finite-measure-preserving actions of two (possibly different) continuous (respectively, discrete) amenable unimodular locally compact second countable groups are orbit equivalent.
The paper under review gives an account of various notions and results about ergodic actions which, though vaguely leading to the rigidity theorem noted above, covers much more than what is directly involved in the proof of the theorem. It is written in a relaxed style and illustrates various ideas in the area. Proofs of many (but not all) well- chosen assertions and also many examples are included.
It may be noted that the proof of rigidity of ergodic actions is motivated by Margulis’ proof of ”superrigidity” of lattices in the same class of groups, which in turn was a major ingredient of his proof of arithmeticity of these lattices. In the later sections of the paper, the present author gives an exposition of the arithmeticity of these lattices. In the later sections of the paper, the present author gives an expostion of the arithmeticity theorem and also certain other more recent results about lattices. The author expects some of these to have appropriate analogues for ergodic actions of the Lie groups in question.

MSC:
22D40 Ergodic theory on groups
28D05 Measure-preserving transformations
22E46 Semisimple Lie groups and their representations
43A05 Measures on groups and semigroups, etc.
22E40 Discrete subgroups of Lie groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, Princeton, N.J., 1963. · Zbl 0099.39103
[2] Louis Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227 – 261. , https://doi.org/10.1090/S0002-9904-1973-13134-9 Louis Auslander, An exposition of the structure of solvmanifolds. II. \?-induced flows, Bull. Amer. Math. Soc. 79 (1973), no. 2, 262 – 285. · Zbl 0265.22016
[3] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0186.33201
[4] Armand Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179 – 188. · Zbl 0094.24901
[5] Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485 – 535. · Zbl 0107.14804
[6] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111 – 164 (French). · Zbl 0143.05901
[7] Jonathan Brezin and Calvin C. Moore, Flows on homogeneous spaces: a new look, Amer. J. Math. 103 (1981), no. 3, 571 – 613. · Zbl 0506.22008
[8] A. Connes, J. Feldman and B. Weiss, Amenable equivalence relations are generated by a single transformation (preprint). · Zbl 0491.28018
[9] C. Delaroche and A. Kirillov, Sur les relations entre l’espace dual d’un groupe et la structure de ses sous-groupes fermés, Seminarie Bourbaki, no. 343, 1967/68. · Zbl 0214.04602
[10] H. A. Dye, On groups of measure preserving transformation. I, Amer. J. Math. 81 (1959), 119 – 159. · Zbl 0087.11501
[11] H. A. Dye, On groups of measure preserving transformations. II, Amer. J. Math. 85 (1963), 551 – 576. · Zbl 0191.42803
[12] Edward G. Effros, Transformation groups and \?*-algebras, Ann. of Math. (2) 81 (1965), 38 – 55. · Zbl 0152.33203
[13] 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. · Zbl 0392.28023
[14] Hans Freudenthal, Topologische Gruppen mit genügend vielen fastperiodischen Funktionen, Ann. of Math. (2) 37 (1936), no. 1, 57 – 77 (German). · Zbl 0013.20202
[15] Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335 – 386. · Zbl 0192.12704
[16] H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477 – 515. · Zbl 0199.27202
[17] Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204 – 256. · Zbl 0347.28016
[18] Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. · Zbl 0174.19001
[19] Yves Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333 – 379 (French). · Zbl 0294.43003
[20] G. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241-260. · JFM 65.0793.02
[21] Eberhard Hopf, Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II, Math. Ann. 117 (1940), 590 – 608 (German). · Zbl 0023.26801
[22] Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), no. 1, 72 – 96. · Zbl 0404.22015
[23] J. W. Jenkins, Growth of connected locally compact groups, J. Functional Analysis 12 (1973), 113 – 127. · Zbl 0247.43001
[24] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71 – 74 (Russian).
[25] Wolfgang Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), no. 1, 19 – 70. · Zbl 0332.46045
[26] George W. Mackey, Ergodic transformation groups with a pure point spectrum, Illinois J. Math. 8 (1964), 593 – 600. · Zbl 0255.22014
[27] G. A. Margulis, Non-uniform lattices in semisimple algebraic groups, in Lie groups and their representations , Wiley, New York. · Zbl 0316.22009
[28] G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 21 – 34 (Russian).
[29] Дискретные подгруппы групп Ли., Издат. ”Мир”, Мосцощ, 1977 (Руссиан). Транслатед фром тхе Енглиш бы О. В. Šварцман; Едитед бы Ѐ. Б. Винберг; Щитх а супплемент ”Аритхметициты оф ирредуцибле латтицес ин семисимпле гроупс оф ранк греатер тхан 1” бы Г. А. Маргулис.
[30] G. A. Margulis, Factor-groups of discrete subgroups, Dokl. Akad. Nauk SSSR 242 (1978), no. 3, 533 – 536 (Russian). · Zbl 0429.20044
[31] G. A. Margulis, Factor groups of discrete subgroups and measure theory, Funktsional. Anal. i Prilozhen. 12 (1978), no. 4, 64 – 76 (Russian).
[32] Calvin C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154 – 178. · Zbl 0148.37902
[33] Calvin C. Moore, Amenable subgroups of semisimple groups and proximal flows, Israel J. Math. 34 (1979), no. 1-2, 121 – 138 (1980). · Zbl 0431.22014
[34] Calvin C. Moore and Robert J. Zimmer, Groups admitting ergodic actions with generalized discrete spectrum, Invent. Math. 51 (1979), no. 2, 171 – 188. · Zbl 0399.22005
[35] G. D. Mostow, Quasi-conformal mappings in \?-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53 – 104. · Zbl 0189.09402
[36] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. · Zbl 0265.53039
[37] D. Ornstein and B. Weiss (to appear).
[38] William Parry, Zero entropy of distal and related transformations, Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967) Benjamin, New York, 1968, pp. 383 – 389. · Zbl 0193.51602
[39] Gopal Prasad, Strong rigidity of \?-rank 1 lattices, Invent. Math. 21 (1973), 255 – 286. · Zbl 0264.22009
[40] Дискретные подгруппы групп Ли., Издат. ”Мир”, Мосцощ, 1977 (Руссиан). Транслатед фром тхе Енглиш бы О. В. Šварцман; Едитед бы Ѐ. Б. Винберг; Щитх а супплемент ”Аритхметициты оф ирредуцибле латтицес ин семисимпле гроупс оф ранк греатер тхан 1” бы Г. А. Маргулис.
[41] Arlan Ramsay, Virtual groups and group actions, Advances in Math. 6 (1971), 253 – 322 (1971). · Zbl 0216.14902
[42] Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147 – 164. · Zbl 0201.36603
[43] Thomas Sherman, A weight theory for unitary representations, Canad. J. Math. 18 (1966), 159 – 168. · Zbl 0136.11601
[44] C. Sutherland, Orbit equivalence: Lectures on Krieger’s theorem, Univ. of Oslo Lecture Notes.
[45] V. S. Varadarajan, Geometry of quantum theory. Vol. II, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. The University Series in Higher Mathematics. · Zbl 0194.28802
[46] P. S. Wang, On isolated points in the dual spaces of locally compact groups, Math. Ann. 218 (1975), no. 1, 19 – 34. · Zbl 0332.22009
[47] Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373 – 409. · Zbl 0334.28015
[48] Robert J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20 (1976), no. 4, 555 – 588. · Zbl 0349.28011
[49] Robert J. Zimmer, Orbit spaces of unitary representations, ergodic theory, and simple Lie groups, Ann. of Math. (2) 106 (1977), no. 3, 573 – 588. · Zbl 0393.22006
[50] Robert J. Zimmer, Uniform subgroups and ergodic actions of exponential Lie groups, Pacific J. Math. 78 (1978), no. 1, 267 – 272. · Zbl 0401.22008
[51] 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. · Zbl 0391.28011
[52] Robert J. Zimmer, Induced and amenable ergodic actions of Lie groups, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 3, 407 – 428. · Zbl 0401.22009
[53] R. J. Zimmer, Algebraic topology of ergodic Lie group action and measurable foliations (preprint).
[54] Robert J. Zimmer, An algebraic group associated to an ergodic diffeomorphism, Compositio Math. 43 (1981), no. 1, 59 – 69. · Zbl 0491.58020
[55] 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
[56] Robert J. Zimmer, Orbit equivalence and rigidity of ergodic actions of Lie groups, Ergodic Theory Dynamical Systems 1 (1981), no. 2, 237 – 253. · Zbl 0485.22013
[57] Robert J. Zimmer, On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups, Amer. J. Math. 103 (1981), no. 5, 937 – 951. · Zbl 0475.22011
[58] Robert J. Zimmer, On the Mostow rigidity theorem and measurable foliations by hyperbolic space, Israel J. Math. 43 (1982), no. 4, 281 – 290. · Zbl 0554.58035
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.