Mathematische Annalen

, Volume 346, Issue 4, pp 969–989 | Cite as

Strongly solid group factors which are not interpolated free group factors

  • Cyril HoudayerEmail author


We give examples of non-amenable infinite conjugacy classes groups Γ with the Haagerup property, weakly amenable with constant Λcb(Γ) = 1, for which we show that the associated II1 factors L(Γ) are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra \({P \subset L(\Gamma)}\) generates an amenable von Neumann algebra. Nevertheless, for these examples of groups Γ, L(Γ) is not isomorphic to any interpolated free group factor L(F t ), for 1 < t ≤  ∞.

Mathematics Subject Classification (2000)

Primary 46L10 46L54 Secondary 22D25 37A20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bożejko M., Picardello M.A.: Weakly amenable groups and amalgamated products. Proc. Am. Math. Soc. 117, 1039–1046 (1993)zbMATHCrossRefGoogle Scholar
  2. 2.
    Brown, N., Ozawa, N.: C*-algebras and finite-dimensional approximations. Grad. Stud. Math., vol. 88. Amer. Math. Soc. Providence, RI (2008)Google Scholar
  3. 3.
    Cherix, P.A., Cowling, M., Jolissaint, P., Julg, P., Valette, A.: Groups with the Haagerup Property. Progress in Mathematics, vol. 197. Birkhäuser Verlag, Basel (2001)Google Scholar
  4. 4.
    Chifan, I., Houdayer, C.: Prime factors and amalgamated free products. arXiv:0805.1566Google Scholar
  5. 5.
    Connes A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cowling M., Haagerup U.: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96, 507–549 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dykema K.: Interpolated free group factors. Pacific J. Math. 163, 123–135 (1994)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Elek G., Szabó E.: On sofic groups. J. Group Theory 9(2), 161–171 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gaboriau D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gaboriau D.: Invariants 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ge L.: Applications of free entropy to finite von Neumann algebras, II. Ann. Math. 147, 143–157 (1998)zbMATHCrossRefGoogle Scholar
  12. 12.
    Gromov M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1, 109–197 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Haagerup U.: An example of non-nuclear C*-algebra which has the metric approximation property. Invent. Math. 50, 279–293 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Houdayer, C.: Construction of type II1 factors with prescribed countable fundamental group. J. Reine Angew Math. (to appear). arXiv:0704.3502Google Scholar
  15. 15.
    Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. arXiv:0805.2998Google Scholar
  16. 16.
    Ioana A., Peterson J., Popa S.: Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200, 85–153 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jolissaint P.: Borel cocycles, approximation properties and relative property T. Ergod. Theory Dyn. Syst. 20, 483–499 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Jung K.: Strongly 1-bounded von Neumann algebras. Geom. Funct. Anal. 17, 1180–1200 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jung K.: A hyperfinite inequality for free entropy dimension. Proc. Am. Math. Soc. 134(7), 2099–2108 (2006)zbMATHCrossRefGoogle Scholar
  20. 20.
    Ozawa N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ozawa, N., Popa, S.: On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. (to appear) arXiv:0706.3623Google Scholar
  22. 22.
    Ozawa, N., Popa, S.: On a class of II1 factors with at most one Cartan subalgebra II. arXiv:0807.4270Google Scholar
  23. 23.
    Peterson J.: L 2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Peterson, J., Thom, A.: Group cocycles and the ring of affiliated operators. arXiv:0708.4327Google Scholar
  25. 25.
    Popa S.: On the superrigidity of malleable actions with spectral gap. J. Am. Math. Soc. 21, 981–1000 (2008)CrossRefGoogle Scholar
  26. 26.
    Popa, S.: Some results and problems in W *-rigidity. Available at
  27. 27.
    Popa S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165, 369–408 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Popa S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)zbMATHCrossRefGoogle Scholar
  29. 29.
    Rădulescu F.: Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index. Invent. Math. 115, 347–389 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Xu Q., Xu Q.: Khintchine type inequalities for reduced free products and applications. J. Reine Angew. Math. 599, 27–59 (2006)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Vaes S.: Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. École Norm. Sup. 41, 743–788 (2008)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Vaes, S.: Rigidity results for Bernoulli actions and their von Neumann algebras (after S. Popa). S’eminaire Bourbaki, exposé, 961. Astérisque 311, 237–294 (2007)MathSciNetGoogle Scholar
  33. 33.
    Voiculescu D.-V.: The analogues of entropy and of Fisher’s information measure in free probability theory, III. Geom. Funct. Anal. 6, 172–199 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Voiculescu, D.-V., Dykema, K.J., Nica, A.: Free random variables. CRM Monograph Series 1. American Mathematical Society, Providence, RI (1992)Google Scholar
  35. 35.
    Weiss B.: Sofic groups and dynamical systems, (Ergodic theory and harmonic analysis, Mumbai, 1999). Sankya Ser. A 62, 350–359 (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.CNRS-ENS LyonLyon cedex 7France

Personalised recommendations