Inventiones mathematicae

, Volume 170, Issue 2, pp 243–295

Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups



We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. \(\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2\), or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and \(\mathcal{V}\) is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. \(\mathcal{V}\) countable discrete, or separable compact), then any \(\mathcal{V}\)-valued measurable cocycle for a measure preserving action \(\Gamma\curvearrowright X\) of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action \(\Gamma\curvearrowright[0,1]^{\Gamma}\)) is cohomologous to a group morphism of Γ into \(\mathcal{V}\). We use the case \(\mathcal{V}\) discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between \(\Gamma\curvearrowright X\) and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.


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  1. 1.
    Adams, S.: Indecomposability of treed equivalence relations. Isr. J. Math. 64, 362–380 (1988)Google Scholar
  2. 2.
    Adams, S.: Indecomposability of equivalence relations generated by word hyperbolic groups. Topology 33, 785–798 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burger, M.: Kazhdan constants for SL(3,ℤ). J. Reine Angew. Math. 413, 36–67 (1991)MATHMathSciNetGoogle Scholar
  4. 4.
    Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., Valette, A.: Groups with Haagerup Property. Birkhäuser, Basel Berlin Boston (2000)Google Scholar
  5. 5.
    Connes, A.: Une classification des facteurs de type III. Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 133–252 (1973)MATHMathSciNetGoogle Scholar
  6. 6.
    Connes, A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Connes, A.: A type II1 factor with countable fundamental group. J. Oper. Theory 4, 151–153 (1980)MATHMathSciNetGoogle Scholar
  8. 8.
    Connes, A., Feldman, J., Weiss, B.: An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dyn. Syst. 1, 431–450 (1981)MATHMathSciNetGoogle Scholar
  9. 9.
    Connes, A., Jones, V.F.R.: A II1 factor with two non-conjugate Cartan subalgebras. Bull. Am. Math. Soc. 6, 211–212 (1982)MATHMathSciNetGoogle Scholar
  10. 10.
    de Cornulier, Y.: Relative Kazhdan property. Ann. Sci. Éc. Norm. Supér., IV. Sér. 39, 301–333 (2006)MATHGoogle Scholar
  11. 11.
    Dixmier, J.: Les C*-Algébres et Leurs Représentations. Gauthier-Villars, Paris (1969)Google Scholar
  12. 12.
    Dixmier, J.: Sous anneaux abéliens maximaux dans les facteurs de type fini. Ann. Math. 59, 279–286 (1954)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Dye, H.: On groups of measure preserving transformations. I. Am. J. Math 81, 119–159 (1959)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dye, H.: On groups of measure preserving transformations. II. Am. J. Math. 85, 551–576 (1963)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras I, II. Trans. Am. Math. Soc. 234, 289–359 (1977)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fernos, T.: Relative property (T) and linear groups. Ann. Inst. Fourier 56, 1767–1804 (2006)MATHMathSciNetGoogle Scholar
  17. 17.
    Fisher, D., Hitchman, T.: Cocycle superrigidity and harmonic maps with infinite dimensional targets. Preprint (2005) (math.DG/0511666)Google Scholar
  18. 18.
    Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. Math. 150, 1059–1081 (1999)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Furman, A.: Orbit equivalence rigidity. Ann. Math. 150, 1083–1108 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Furman, A.: Outer automorphism groups of some ergodic equivalence relations. Comment. Math. Helv. 80, 157–196 (2005)MATHMathSciNetGoogle Scholar
  21. 21.
    Furman, A.: On Popa’s cocycle superrigidity theorem. Preprint (2006)Google Scholar
  22. 22.
    Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions. J. Anal. Math. 31, 204–256 (1977)MATHMathSciNetGoogle Scholar
  23. 23.
    Gaboriau, D.: Cout des rélations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Gaboriau, D.: Invariants ℓ2 de rélations d’équivalence et de groupes. Publ. Math., Inst. Hautes Étud. Sci. 95, 93–150 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gefter, S.L.: On cohomologies of ergodic actions of a T-group on homogeneous spaces of a compact Lie group. In: Operators in Functional Spaces and Questions of Function Theory, pp. 77–83. Collect. Sci. Works, Kiev (1987). (Russian)Google Scholar
  26. 26.
    Gefter, S.L., Golodets, V.Y.: Fundamental groups for ergodic actions and actions with unit fundamental groups. Publ. Res. Inst. Math. Sci. 6, 821–847 (1988)MathSciNetGoogle Scholar
  27. 27.
    de la Harpe, P., Valette, A.: La propriété T de Kazhdan pour les groupes localement compacts. Astérisque, vol. 175. Soc. Math. de France (1989)Google Scholar
  28. 28.
    Hjorth, G., Kechris, A.: Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations. Mem. Am. Math. Soc., vol. 177(833) (2005)Google Scholar
  29. 29.
    Ioana, A., Peterson, J., Popa, S.: Amalgamated free products of w-rigid factors and calculation of their symmetry groups. To appear in Acta Math. (math.OA/0505589)Google Scholar
  30. 30.
    Jackson, S., Kechris, A., Hjorth, G.: Countable Borel equivalence relations. J. Math. Log. 1, 1–80 (2002)CrossRefGoogle Scholar
  31. 31.
    Jolissaint, P.: On property (T) for pairs of topological groups. Enseign. Math., II. Sér. 51, 31–45 (2005)MATHMathSciNetGoogle Scholar
  32. 32.
    Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Kadison, R.V., Singer, I.M.: Some remarks on representations of connected groups. Proc. Nat. Acad. Sci. 38, 419–423 (1952)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kazhdan, D.: Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1, 63–65 (1967)MATHCrossRefGoogle Scholar
  35. 35.
    Margulis, G.: Finitely-additive invariant measures on Euclidian spaces. Ergodic Theory Dyn. Syst. 2, 383–396 (1982)MATHMathSciNetGoogle Scholar
  36. 36.
    Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differ. Geom. 67, 395–455 (2004)MATHMathSciNetGoogle Scholar
  37. 37.
    Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. 164, 825–878 (2006)MathSciNetMATHGoogle Scholar
  38. 38.
    Murray, F., von Neumann, J.: On rings of operators. Ann. Math. 37, 116–229 (1936)CrossRefGoogle Scholar
  39. 39.
    Murray, F., von Neumann, J.: Rings of operators IV. Ann. Math. 44, 716–808 (1943)CrossRefGoogle Scholar
  40. 40.
    von Neumann, J., Segal, I.E.: A theorem on unitary representations of semisimple Lie groups. Ann. Math. 52, 509–516 (1950)CrossRefGoogle Scholar
  41. 41.
    Ornstein, D., Weiss, B.: Ergodic theory of amenable group actions I. The Rohlin lemma. Bull. Am. Math. Soc. (1) 2, 161–164 (1980)MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Popa, S.: Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230, 273–328 (2006)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165, 369–408 (2006) (math.OA/0305306)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups II. Invent. Math. 165, 409–452 (2006) (math.OA/0407137)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Popa, S.: Some computations of 1-cohomology groups and construction of non orbit equivalent actions. J. Inst. Math. Jussieu 5, 309–332 (2006) (math.OA/0407199)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Popa, S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–889 (2006) (math.OA/0209310)MATHCrossRefGoogle Scholar
  47. 47.
    Popa, S.: Classification of Subfactors and of Their Endomorphisms. CBMS Lect. Notes, vol. 86. Am. Math. Soc. (1995)Google Scholar
  48. 48.
    Popa, S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111, 375–405 (1993)MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Popa, S.: Correspondences. INCREST preprint (1986) (unpublished, Scholar
  50. 50.
    Popa, S., Sasyk, R.: On the cohomology of Bernoulli actions. Ergodic Theory Dyn. Syst. 27, 241–251 (2007) (math.OA/0310211)MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Popa, S., Vaes, S.: Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups. Preprint (2006) (math.OA/0605456)Google Scholar
  52. 52.
    Shalom, Y.: Measurable group theory. In: Proceedings of the 2004 European Congress of Mathematics (Stockholm 2004), pp. 391–423. EMS Publishing House, Zürich (2004)Google Scholar
  53. 53.
    Singer, I.M.: Automorphisms of finite factors. Am. J. Math. 77, 117–133 (1955)MATHCrossRefGoogle Scholar
  54. 54.
    Thomas, S.: Popa’s superrigidity and countable Borel equivalence relations. To appearGoogle Scholar
  55. 55.
    Vaes, S.: Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). Séminaire Bourbaki, exposé, vol. 961. To appear in Astérisque. (math.OA/0605456)Google Scholar
  56. 56.
    Valette, A.: Group pairs with relative property (T) from arithmetic lattices. Geom. Dedicata 112, 183–196 (2005)MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Zimmer, R.: Strong rigidity for ergodic actions of seimisimple Lie groups. Ann. Math. 112, 511–529 (1980)CrossRefMathSciNetGoogle Scholar
  58. 58.
    Zimmer, R.: Ergodic Theory and Semisimple Groups. Birkhäuser, Boston (1984)MATHGoogle Scholar
  59. 59.
    Zimmer, R.: Extensions of ergodic group actions. Ill. J. Math. 20, 373–409 (1976)MATHMathSciNetGoogle Scholar
  60. 60.
    Zimmer, R.: Superrigidity, Ratner’s theorem and the fundamental group. Isr. J. Math. 74, 199–207 (1991)MathSciNetMATHGoogle Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Math.Dept., UCLAUniversity of CaliforniaLos AngelesUSA

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