The Structure of Borel Equivalence Relations in Polish Spaces

  • Alexander S. Kechris
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 26)


An exposition of recent work on Borel equivalence relations in Polish spaces is presented. This includes a general Glimm-Effros dichotomy for Borel equivalence relations and a study of countable Borel equivalence relations and their classification into subclasses such as smooth, hyperfinite, amenable, treeable etc.


Equivalence Relation Polish Space Continuum Hypothesis Countable Group Jordan Canonical Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Adams, Trees and amenable equivalence relations, Erg. Theory and Dyn. Systems 10 (1990), 1–14.MATHGoogle Scholar
  2. 2.
    S. Adams, Indecomposability of treed equivalence relations, Israel J. Math 64(3) (1988), 362–380.MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Adams and R. Spatzier, Kazhdan groups, cocycles and trees, Amer. J. Math 112 (1990), 271–287.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. Burgess, A selection theorem for group actions, Pac. J. Math. 80(2) (1979), 333–336.MathSciNetMATHGoogle Scholar
  5. 5.
    A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Erg. Theory and Dyn. Systems 1 (1981), 431–450.MathSciNetMATHGoogle Scholar
  6. 6.
    R. Dellacherie and P-A. Meyer, Therie discrete du potentiel, Hermann, 1983.Google Scholar
  7. 7.
    R. Dougherty, S. Jackson and A. Kechris, The structure of equivalence relations on Polish spaces, I; An extension of the Glimm-Effros dichotomy, circulated notes, March 1989.Google Scholar
  8. 8.
    R. Dougherty, S. Jackson and A. Kechris, The structure of equivalence relations on Polish spaces, II; Countable equivalence relations or descriptive dynamics, circulated notes, March 1989.Google Scholar
  9. 9.
    E. Effiros, Transformation groups and C*-algebras, Ann. of Math. 81(1) (1965), 38–55.MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Effiros, Polish transformation groups and classification problems, General Topology and Modern Analysis, Rao and McAuley, eds. (1980), Academic Press, 217–227.Google Scholar
  11. 11.
    J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras, I, Trans. Amer. Math. Soc. 234(2) (1977), 289–324.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    M. Foreman, A Dilworth decomposition theorem for X-Suslin quasi-orderings of E, in “Logic, Methodology and Philosophy of Science VIII”, North Holland, 1989, 223–244, 223–244.CrossRefGoogle Scholar
  13. 13.
    H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, J. Symb. Logic 54(3) (1989), 894–914.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    J. Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124–138.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    A. Godefroy, Some remarks on Suslin sections, Fund. Math. LXXXIV (1986), 159–167.MathSciNetGoogle Scholar
  16. 16.
    L. Harrington, A. Kechris and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3(4) (1990), 903–928.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    L. Harrington, D. Marker and S. Shelah, Borel orderings, Trans. Amer. Math. Soc. 310(1) (1988), 293–302.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    L. Harrington and R. Sami, Equivalence relations, projective and beyond, Logic Colloq. 78 (IJorth Holland, 1979), 247–264.MathSciNetGoogle Scholar
  19. 19.
    K. Kada, A Borel version of Dilworth’s theorem, (to appear).Google Scholar
  20. 20.
    Y. Katznelson and B. Weiss, The construction of quasi-invariant measures, Israel J. Math. 12 (1972), 1–4.MathSciNetMATHGoogle Scholar
  21. 21.
    A. Kechris, Amenable equivalence relations and Turing degrees, J. Symb. Logic (to appear).Google Scholar
  22. 22.
    W. Krieger, On Borel automorphisms and their quasi-invariant measures, Math. Z. 151 (1976), 19–24.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    A. Louveau, Two results on Borel orders, J. Symb. Logic 34(3) (1989), 865–874.MATHGoogle Scholar
  24. 24.
    A. Louveau and J. Saint Raymond, On the quasi-ordering of Borel linear orders under embeddability, J. Symb. Logic 55(2) (1990), 537–560.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    D. Mauldin and S. Ulam, Mathematical problems and games, Adv. in Appl. Math 8 (1987), 281–344.MathSciNetMATHGoogle Scholar
  26. 26.
    P. Muhly, K. Saito and B. Solel, Coordinates for triangular operator algebras, Ann. of Math. 127 (1988), 245–278.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    S. Shelah and B. Weiss, Measurable recurrence and quasi-invariant measures, Israel J. Math. 43 (1982), 154–160.MATHGoogle Scholar
  28. 28.
    J. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), 1–28.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    T. Slaman and J. Steel, Definable functions on degrees, in “Cabal Seminar 81–85”, Lecture Notes in Math. 1333, Springer-Verlag, 1988, 37–55.Google Scholar
  30. 30.
    J. Steel, Long games, in “Cabal Seminar 81–85”, Lecture notes in Math. 1333, Springer-Verlag, 1988, 56–97.Google Scholar
  31. 31.
    D. Sullivan, B. Weiss and J.D.M. Wright, Generic dynamics and monotone complete C*-algebras, Trans. Amer. Math. Soc. 295(2) (1986), 795–809.MathSciNetMATHGoogle Scholar
  32. 32.
    A. Vershik, The action of PSL(2, Z) on M1 is approximate, Uspekhi Mat. Nauk. 33(1) (1978), 209–210 (in Russian).MathSciNetMATHGoogle Scholar
  33. 33.
    S. Wagon, The Banach-Tarski Paradox, Cambridge Univ. Press, 1985.MATHGoogle Scholar
  34. 34.
    Weiss, Measurable dynamics, Cont. Math. 26 (1984), 395–421.MATHGoogle Scholar
  35. 35.
    R. Zimmer, Hyperfinite factors and amenable ergodic actions, Inv. Math 41 (1977), 23–31.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Alexander S. Kechris
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations