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Ergodic Subequivalence Relations Induced by a Bernoulli Action

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Abstract

Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\), where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\), there exists a partition {X i }i≥0 of [0, 1]Γ into \({\mathcal{R}}\)-invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

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References

  1. AM

    M. Abért, N. Nikolov, The rank gradient from a combinatorial point of view, preprint; arXiv:math/0701925

  2. C

    Connes A.: Outer conjugacy classes of automorphisms of factors. Ann. Ec. Norm. Sup. 8, 383–419 (1975)

    MATH  MathSciNet  Google Scholar 

  3. CFW

    Connes A., Feldman J., Weiss B.: An amenable equivalence relations is generated by a single transformation. Ergodic Th. Dynam. Sys. 1, 431–450 (1981)

    MATH  MathSciNet  Google Scholar 

  4. CW

    Connes A., Weiss B.: Property (T) and asymptotically invariant sequences. Israel J. Math. 37, 209–210 (1980)

    MATH  Article  MathSciNet  Google Scholar 

  5. D

    Dye H.: On groups of measure preserving transformations II. Amer. J. Math. 85, 551–576 (1963)

    MATH  Article  MathSciNet  Google Scholar 

  6. FM

    Feldman J., Moore C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras, II. Trans. Am. Math. Soc. 234, 325–359 (1977)

    MATH  Article  MathSciNet  Google Scholar 

  7. Fu

    A. Furman, A survey of measured group theory, preprint; arXiv:0901.0678

  8. GL

    D. Gaboriau, R. Lyons, A-measurable-group-theoretic solution to von Neumann’s problem, Invent. Math., to appear; arXiv:0711.1643

  9. HK

    Hjorth G., Kechris A.: Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations. Memoirs of the Amer. Math. Soc. 177, 833 (2005)

    MathSciNet  Google Scholar 

  10. I

    Ioana A.: Rigidity results for wreath product II1 factors. Journal of Functional Analysis 252, 763–791 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  11. IPP

    Ioana A., Peterson J., Popa S.: Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200(1), 85–153 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  12. JS

    Jones V.F.R., Schmidt K.: Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109, 91–114 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  13. KT

    Kechris A., Tsankov T.: Amenable actions and almost invariant sets. Proc. Amer. Math. Soc. 136(2), 687–697 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  14. LPS

    Lyons R., Peres Y., Schramm O.: Minimal spanning forests. Ann. Probab. 27, 1665–1692 (2006)

    Article  MathSciNet  Google Scholar 

  15. MN

    Murray F., von Neumann J.: Rings of operators, IV. Ann. Math. 44, 716–808 (1943)

    Article  Google Scholar 

  16. OW

    Ornstein D.S., Weiss B.: Ergodic theory of amenable group actions. Bull. Amer. Math. Soc. (N.S.) 2, 161–164 (1980)

    MATH  Article  MathSciNet  Google Scholar 

  17. Oz1

    Ozawa N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  18. Oz2

    N. Ozawa, A Kurosh type theorem for type II1 factors, Int. Math. Res. Not. 2006 (2006), article ID 97560.

  19. Oz3

    N. Ozawa, An example of a solid von Neumann algebra, Hokkaido Math. J., to appear; arXiv:0804.0288

  20. P

    Peterson J.: L2-rigidity in von Neumann algebras. Invent. Math. 175(2), 417–433 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  21. Po1

    Popa S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165, 369–408 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  22. Po2

    Popa S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups II. Invent. Math. 165, 409–451 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  23. Po3

    Popa S.: Cocycle and orbit equivalence superrigidity for Bernoulli actions of Kazhdan groups. Invent. Math 170, 243–295 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  24. Po4

    Popa S.: On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21(4), 981–1000 (2008)

    Article  MathSciNet  Google Scholar 

  25. Po5

    S. Popa, Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians I, Eur. Math. Soc. Zürich (2007) 445–477.

  26. Po6

    S. Popa, On Ozawa’s property for free group factors, Int. Math. Res. Not. 2007 (2007), article ID rnm036.

  27. S

    Schmidt K.: Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions. Ergod. Th. Dynam. Sys. 1, 223–236 (1981)

    MATH  Article  Google Scholar 

  28. Sh

    Y. Shalom, Measurable group theory, European Congress of Mathematics, Eur. Math. Soc., Zürich (2005), 391–423.

  29. T

    Timár Á.: Ends in minimal spanning forests. Ann. Probab. 34, 865–869 (2006)

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Adrian Ioana.

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The second author was supported by a Clay Research Fellowship.

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Chifan, I., Ioana, A. Ergodic Subequivalence Relations Induced by a Bernoulli Action. Geom. Funct. Anal. 20, 53–67 (2010). https://doi.org/10.1007/s00039-010-0058-7

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Keywords and phrases

  • Bernoulli action
  • deformation/rigidity
  • ergodic subequivalence relation
  • malleable
  • strongly ergodic

2010 Mathematics Subject Classification

  • 37A20
  • 37A15