Skip to main content
SpringerLink
Log in
Book cover

Harmonic Analysis and Group Representation pp 424–484Cite as

Ergodic Theory, Group Representations, and Rigidity

  • Chapter

  • 1229 Accesses

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 82))

Abstract

In these lectures we discuss some topics concerning the relationship of ergodic theory, representation theory, and the structure of Lie groups and their discrete subgroups.

In studying the representation theory of groups, the assumption of compactness on the group essentially allows one to reduce to a finite dimensional situation, in which case one often can obtain complete information. For non-compact groups, of course, no such reduction is possible and the situation is much more complex. When studying general actions of groups, a somewhat similar situation arises. In the compact case every orbit will be closed, the space of orbits will have a reasonable structure, and one can often find nice (with respect to the action) neighborhoods of orbits. A large amount of information about actions of finite and compact groups has been obtained by topological methods. However, once again, if the compactness assumption on the group is dropped, one faces many additional problems. In particular, one can have orbits which are dense (for example, the irrational flow on the torus) and the orbit space may be so badly behaved as to have no continuous functions but constants. Furthermore, moving from a point to a nearby point may produce an orbit which doesn't follow closely to the original orbit. If one wishes to deal with actions in the non-compact case, this phenomenon of complicated orbit structure must be faced. For many actions, e.g., diffentiable actions on manifolds, there are natural measures that behave well with respect to the action. A significant part of ergodic theory is the study of group actions on measure spaces. In particular, ergodic theory aims to understand the phenomenon of bad orbit structure in the presence of a measure.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Auslander, L. Green, F. Hahn, Flows on Homogeneous Spaces, Annals of Math. Studies, no.53, Princeton, 1963

    MATH  Google Scholar 

  2. L. Auslander, An Exposition of the Structure of Solvmanifolds, Bull. Amer. Math. Soc, 79(1973), 227–285.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Borel, Linear Algebraic Groups, Benjamin, New York, 1969.

    MATH  Google Scholar 

  4. A. Borel, Density Properties for Certain Subgroups of Semisimple Lie Groups Without Compact Factors, Annals of Math., 72(1960), 179–188.

    Article  MathSciNet  Google Scholar 

  5. A. Borel, Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, Annals of Math., 75(1962), 485–535.

    Article  MathSciNet  Google Scholar 

  6. A. Borel, J .P. Serre, Theorèmes de Finitude en Cohomologie Galoisienne, Comm. Math. Helv., 39(1964), 111–164.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Brezin, CC. Moore, Flows on Homogeneous Spaces: A New Look, preprint.

    Google Scholar 

  8. A. Connes. J. Feldman, B̀. Weiss, Amenable equivalence relations are generated by a single transformation, preprint.

    Google Scholar 

  9. C. Delaroche, A Kirillov, Sur Les Relations Entre L'Espace Dual d'un Groupe et la Structure de ses Sous-Groupes Fermés, Seminaire Bourbaki, no.343, 1967/68.

    Google Scholar 

  10. H.A. Dye, On Groups of Measure Preserving Transformations, I, Amer. J. Math., 81(1959), 119–159.

    Article  MathSciNet  MATH  Google Scholar 

  11. H.A. Dye, On Groups of Measure Preserving Transformations, II, Amer. J. Math., 85(1963), 551–576.

    Article  MathSciNet  MATH  Google Scholar 

  12. E.G. Effros, Transformation Groups and C*-Algebras, Annals of Math., 81(1965), 38–55.

    Article  MathSciNet  Google Scholar 

  13. J. Feldman, P. Hahn, C .C. Moore, Orbit Structure and Countable Sections for Actions of Continuous Groups, Advances in Math., 28(1978), 186–230.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Freudenthal, Topologische Gruppen mit Genugend Vielen Fastperiodishen Funktionen, Annals of Math., 37(1936), 57–77.

    Article  MathSciNet  Google Scholar 

  15. H. Furstenberg, A Poisson Formula for Semi simple Lie Groups, Annals of Math., 77(1963), 335–383.

    Article  MathSciNet  Google Scholar 

  16. H. Furstenberg, The Structure of Distal Flows, Amer. J. Math., 85(1963), 477–515.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Furstenberg, Ergodic Behavior of Diagonal Measures and a Theorem of Szemeredi on Arithmetic Progressions, J. Analyse Math., 31(1977), 204–256.

    Article  MathSciNet  MATH  Google Scholar 

  18. F.P. Greenleaf, Invariant Means on Toplogical Groups, Van Nostrand, New York, 1969.

    Google Scholar 

  19. Y. Guivarc'h, Croissance Polynomiale et Periodes des Fonctions Harmonique, Bull. Math. Soc. France, 101(1973), 333–379

    MathSciNet  MATH  Google Scholar 

  20. G. Hedlund, The Dynamics of Geodesic Flows, Bull. Amer. Math. Soc, 45(1939), 241–260.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Hopf, Statistik der Lösungen geodätischer probleme vom unstabilen typus, Math. Ann. 117(1940), 590–608.

    Article  MathSciNet  Google Scholar 

  22. R. Howe and C .C. Moore, Asymptotic Properties of Unitary Representations, J. Func, Anal .,32(1979), 72–96.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.W. Jenkins, Growth of Connected Locally Compact Groups, J. Funct. Anal ., 12(1973), 113–127.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Kazhdan, Connection of the Dual Space of a Group with the Structure of its Closed Subgroups, Funct. Anal. Appl., 1(1967), 63–65.

    Article  MATH  Google Scholar 

  25. W. Krieger, On Ergodic Flows and the Isomorphism of Factors, Math. Ann. 223(1976), 19–70.

    Article  MathSciNet  MATH  Google Scholar 

  26. G.W. Mackey, Ergodic Transformation Groups with a Pure Point Spectrum, Illinois J. Math., 8(1964), 593–600.

    MathSciNet  MATH  Google Scholar 

  27. G.A. Margulis, Non-uniform lattices in Semi simple Algebraic Groups, in Lie Groups and their Representations, ed. I .M. Gelfand, Wiley, New York.

    Google Scholar 

  28. G.A. Margulis, Discrete Groups of Motions of Manifolds of Non-Positive Curvature, Amer. Math. Soc. Translations, 109(1977), 33–45.

    MATH  Google Scholar 

  29. G.A. Margulis, Arithmeticity of Irreducible Lattices in Semisimple Groups of Rank Greater than 1, Appendix to Russian Translation of M. Ragunathan, Discrete Subgroups of Lie Groups, Mir, Moscow, 1977(in Russian).

    Google Scholar 

  30. G.A. Margulis, Factor Groups of Discrete Subgroups, Soviet Math. Dokl. 19(1978), 1145–1149.

    MATH  Google Scholar 

  31. G.A. Margulis, Quotient Groups of Discrete Subgroups and Measure Theory, Funct. Anal. Appl., 12(1978), 295–305.

    Article  MathSciNet  MATH  Google Scholar 

  32. C.C. Moore, Ergodicity of Flows on Homogeneous Spaces, Amer. J. Math., 88(1966), 154–178.

    Article  MathSciNet  MATH  Google Scholar 

  33. C.C. Moore, Amenable Subgroups of Semisimple Groups and Proximal Flows, Israel J. Math., 34(1979), 121–138.

    Article  MathSciNet  MATH  Google Scholar 

  34. C.C. Moore and R.J. Zimmer, Groups Admitting Ergodic Actions with Generalized Discrete Spectrum, Invent. Math., 51(1979), 171–188.

    Article  MathSciNet  MATH  Google Scholar 

  35. G.D. Mostow, Quasi-Conformal Mappings in n - Space and the Rigidity -of Hyperbolic Space Forms, Publ . Math. I.H.E.S., (1967), 53–104.

    Google Scholar 

  36. G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, no.78, Princeton Univ. Press, Princeton, N.J. 1973.

    MATH  Google Scholar 

  37. D. Ornstein and B. Weiss, to appear.

    Google Scholar 

  38. W. Parry, Zero Entropy of Distal and Related Transformations, in Topological Dynamics, eds. J. Auslander, W. Gottschalk, Benjamin, New York, 1968.

    Google Scholar 

  39. G. Prasad, Strong Rigidity of Q-rank 1 Lattices, Invent. Math., 21(1973), 255–286.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Ragunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, New York, 1972.

    Google Scholar 

  41. A. Ramsay, Virtual Groups and Group Actions, Advances in Math., 6(1971), 253–322.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Selberg, On Discontinuous Groups in Higher Dimensional Symmetric Spaces, Int. Colloquium on Function Theory, Tata Institute, Bombay, 1960.

    Google Scholar 

  43. T. Sherman, A Weight Theory for Unitary Representations, Canadian J. Math., 18(1966), 159–168.

    Article  MathSciNet  MATH  Google Scholar 

  44. C. Sutherland, Orbit Equivalence: Lectures on Kriegers Theorem, University of Oslo Lecture Notes.

    Google Scholar 

  45. V.S. Varadarajan, Geometry of Quantum Theory, vol. II., Van Nostrand, Princeton, N.J. 1970.

    MATH  Google Scholar 

  46. S.P. Wang, On Isolated Points in the Dual Spaces of Locally Compact Groups, Math. Ann., 218(1975), 19–34.

    Article  MathSciNet  MATH  Google Scholar 

  47. R.J. Zimmer, Extensions of Ergodic Group Actions, Illinois J. Math., 20(1976), 373–409.

    MathSciNet  MATH  Google Scholar 

  48. R.J. Zimmer, Ergodic Actions with Generalized Discrete Spectrum, Illinois J. Math., 20(1976), 555–588.

    MathSciNet  MATH  Google Scholar 

  49. R.J. Zimmer, Orbit Spaces of Unitary Representations, Ergodic Theory, and Simple Lie Groups, Annals of Math., 106(1977), 573–588.

    Article  MathSciNet  Google Scholar 

  50. R.J. Zimmer, Uniform Subgroups and Ergodic Actions of Exponential Lie Groups, Pac. J. Math., 78(1978), 267–272.

    MathSciNet  MATH  Google Scholar 

  51. R.J. Zimmer, Amenable Ergodic Group Actions and an Application to Poisson Boundaries of Random Walks, J. Funct. Anal., 27(1978), 350–372.

    Article  MathSciNet  MATH  Google Scholar 

  52. R.J. Zimmer, Induced and Amenable Ergodic Actions of Lie Groups, Ann. Sci. Ec. Norm. Sup., 11(1978), 407–428.

    MathSciNet  MATH  Google Scholar 

  53. R.J. Zimmer, Algebraic Topology of Ergodic Lie Group Action and Measurable Foliations, preprint.

    Google Scholar 

  54. R.J. Zimmer, An Algebraic Group Associated to an Ergodic Diffeomorphism, Comp. Math., to appear.

    Google Scholar 

  55. R.J. Zimmer, Strong Rigidity for Ergodic Actions of Semisimple Lie Groups, Annals of Math., to appear.

    Google Scholar 

  56. R.J. Zimmer, Orbit Equivalence and Rigidity of Ergodic Actions of Lie Groups, preprint.

    Google Scholar 

  57. R.J. Zimmer, On the Cohomology of Ergodic Actions of Semisimple lie Groups and Discrete subgroups, Amer. J. Math., to appear.

    Google Scholar 

  58. R.J. Zimmer, On the Mostow Rigidity Theorem and Measurable Foliations by Hyperbolic Space, preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, USA

    Robert J. Zimmer

Authors
  1. Robert J. Zimmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

A. Figà Talamanca

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Zimmer, R.J. (2010). Ergodic Theory, Group Representations, and Rigidity. In: Talamanca, A.F. (eds) Harmonic Analysis and Group Representation. C.I.M.E. Summer Schools, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11117-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation