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Maximal von Neumann subalgebras arising from maximal subgroups

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Abstract

Ge (2003) asked the question whether LF∞ can be embedded in to LF2 as a maximal subfactor. We answer it affirmatively in three different approaches, all containing the same key ingredient: the existence of maximal subgroups with infinite index. We also show that point stabilizer subgroups for every faithful, 4-transitive action on an infinite set give rise to maximal von Neumann subalgebras. By combining this with the known results on constructing faithful, highly transitive actions, we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.

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References

  1. Arzhantseva G, Minasyan A, Osin D. The SQ-universality and residual properties of relatively hyperbolic groups. J Algebra, 2007, 315: 165–177

    Article  MathSciNet  MATH  Google Scholar 

  2. Bédos E. On actions of amenable groups on II1-factors. J Funct Anal, 1990, 91: 404–414

    Article  MathSciNet  MATH  Google Scholar 

  3. Boutonnet R, Carderi A. Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom Funct Anal, 2015, 25: 1688–1705

    Article  MathSciNet  MATH  Google Scholar 

  4. Boutonnet R, Carderi A. Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups. Math Ann, 2017, 367: 1199–1216

    Article  MathSciNet  MATH  Google Scholar 

  5. Cameron J. Structure results for normalizers of II1 factors. Internat J Math, 2011, 22: 947–979

    Article  MathSciNet  MATH  Google Scholar 

  6. Cameron J, Christensen E, Sinclair A, et al. Kadison-Kastler stable factors. Duke Math J, 2014, 163: 2639–2686

    Article  MathSciNet  MATH  Google Scholar 

  7. Cameron J, Fang J, Ravichandran M, et al. The radial masa in a free group factor is maximal injective. J Lond Math Soc (2), 2010, 82: 787–809

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaynikov V V. Properties of hyperbolic groups: Free normal subgroups, quasiconvex subgroups and actions of maximal growth. PhD Thesis. Nashville: Vanderbilt University, 2012

    Google Scholar 

  9. Chifan I, Das S. Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces. Math Ann, 2020, 378: 907–950

    Article  MathSciNet  MATH  Google Scholar 

  10. Choda H. A Galois correspondence in a von Neumann algebra. Tohoku Math J (2), 1978, 30: 491–504

    Article  MathSciNet  MATH  Google Scholar 

  11. Choda M. Some relations of II1-factors on free groups. Math Japonica, 1977, 22: 383–394

    MathSciNet  MATH  Google Scholar 

  12. Connes A, Jones V. Property T for von Neumann algebras. Bull Lond Math Soc, 1985, 17: 57–62

    Article  MathSciNet  MATH  Google Scholar 

  13. Dahmani F, Guirardel V, Osin D. Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces. Memoirs of the American Mathematical Society, vol. 245. Providence: Amer Math Soc, 2017

    Book  MATH  Google Scholar 

  14. de Cornulier Y. Finitely presented wreath products and double coset decompositions. Geom Dedicata, 2006, 122: 89–108

    Article  MathSciNet  MATH  Google Scholar 

  15. Dixon J D. Most finitely generated permutation groups are free. Bull Lond Math Soc, 1990, 22: 222–226

    Article  MathSciNet  MATH  Google Scholar 

  16. Dykema K. Free products of hyperfinite von Neumann algebras and free dimension. Duke Math J, 1993, 69: 97–119

    Article  MathSciNet  MATH  Google Scholar 

  17. Dykema K. Interpolated free group factors. Pacific J Math, 1994, 163: 123–135

    Article  MathSciNet  MATH  Google Scholar 

  18. Fima P, Moon S, Stalder Y. Highly transitive actions of groups acting on trees. Proc Amer Math Soc, 2015, 143: 5083–5095

    Article  MathSciNet  MATH  Google Scholar 

  19. Francoeur D, Garrido A. Maximal subroups of groups of intermediate growth. Adv Math, 2018, 340: 1067–1107

    Article  MathSciNet  MATH  Google Scholar 

  20. Garion S, Glasner Y. Highly transitive actions of Out(Fn). Groups Geom Dyn, 2013, 7: 357–376

    Article  MathSciNet  MATH  Google Scholar 

  21. Ge L. On maximal injective subalgebras of factors. Adv Math, 1996, 118: 34–70

    Article  MathSciNet  MATH  Google Scholar 

  22. Ge L. On “Problems on von Neumann algebras by R. Kadison, 1967”. Acta Math Sin (Engl Ser), 2003, 19: 619–624

    Article  MathSciNet  MATH  Google Scholar 

  23. Ge L, Kadison R. On tensor products of von Neumann algebras. Invent Math, 1996, 123: 453–466

    Article  MathSciNet  MATH  Google Scholar 

  24. Ge L, Popa S. On some decomposition properties of factors of type II1. Duke Math J, 1998, 94: 79–101

    Article  MathSciNet  MATH  Google Scholar 

  25. Gelander T, Glasner Y. Countable primitive groups. Geom Funct Anal, 2008, 17: 1479–1523

    Article  MathSciNet  MATH  Google Scholar 

  26. Gelander T, Glasner Y, Soĭfer G. Maximal subgroups of countable groups, a survey. arXiv:1909.09361, 2019

  27. Gelander T, Meiri C. Maximal subgroups of SL(n, ℤ). Transform Groups, 2016, 21: 1063–1078

    Article  MathSciNet  MATH  Google Scholar 

  28. Glasner Y, Gulko D. Sharply 2-transitive linear groups. In: International Mathematics Research Notices. IMRN. Cary: Oxford University Press, 2014, 2691–2701

    Google Scholar 

  29. Golan G, Sapir M. On subgroups of R. Thompson’s group F. Trans Amer Math Soc, 2017, 369: 8857–8878

    Article  MathSciNet  MATH  Google Scholar 

  30. Gunhouse S V. Highly transitive representations of free products on the natural numbers. Arch Math (Basel), 1992, 58: 435–443

    Article  MathSciNet  MATH  Google Scholar 

  31. Hall M. Coset representations in free groups. Trans Amer Math Soc, 1949, 67: 421–432

    Article  MathSciNet  MATH  Google Scholar 

  32. Hamana M. Injective envelopes of operator systems. Publ Res Inst Math Sci, 1979, 15: 773–785

    Article  MathSciNet  MATH  Google Scholar 

  33. Hickin K. Highly transitive Jordan representations of free products. J Lond Math Soc (2), 1992, 46: 81–91

    Article  MathSciNet  MATH  Google Scholar 

  34. Houdayer C. A class of II1 factors with an exotic abelian maximal amenable subalgebra. Trans Amer Math Soc, 2014, 366: 3693–3707

    Article  MathSciNet  MATH  Google Scholar 

  35. Hull M, Osin D. Transitivity degrees of countable groups and acylindrical hyperbolicity. Israel J Math, 2016, 216: 307–353

    Article  MathSciNet  MATH  Google Scholar 

  36. Jiang Y, Skalski A. Maximal subgroups and von Neumann subalgebras with the Haagerup property. Group Geom Dyn, 2021, in press

  37. Juschenko K, Monod N. Cantor systems, piecewise translations and simple amenable groups. Ann of Math (2), 2013, 178: 775–787

    Article  MathSciNet  MATH  Google Scholar 

  38. Kadison R V, Ringrose J R. Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory. New York: Academic Press, 1983

    MATH  Google Scholar 

  39. Kadison R V, Ringrose J R. Fundamentals of the Theory of Operator Algebras. Volume II: Advanced Theory. Orlando: Academic Press, 1986

    MATH  Google Scholar 

  40. Kitroser D. Highly-transitive actions of surface groups. Proc Amer Math Soc, 2012, 140: 3365–3375

    Article  MathSciNet  MATH  Google Scholar 

  41. Le Boudec A, Matte Bon N. Triple transitivity and non-free actions in dimension one. J Lond Math Soc, 2021, in press

  42. Longo R. Simple injective subfactors. Adv Math, 1987, 63: 152–171

    Article  MathSciNet  MATH  Google Scholar 

  43. Margulis G A, Soĭfer G A. A criterion for the existence of maximal subgroups of infinite index in a finitely generated linear group. Dokl Akad Nauk SSSR, 1977, 234: 1261–1264

    MathSciNet  Google Scholar 

  44. Margulis G A, Soĭfer G A. Non-free maximal subgroups of infinite index of the group SLn(Z). Uspekhi Mat Nauk, 1979, 34: 203–204

    MathSciNet  MATH  Google Scholar 

  45. Margulis G A, Soĭfer G A. Maximal subgroups of infinite index in finitely generated linear groups. J Algebra, 1981, 69: 1–23

    Article  MathSciNet  MATH  Google Scholar 

  46. Matui H. Topological full groups of one-sided shifts of finite type. J Reine Angew Math, 2015, 705: 35–84

    MathSciNet  MATH  Google Scholar 

  47. McDonough T P. A permutation representation of a free group. Q J Math, 1977, 28: 353–356

    Article  MathSciNet  MATH  Google Scholar 

  48. Moon S, Stalder Y. Highly transitive actions of free products. Algebr Geom Topol, 2013, 13: 589–607

    Article  MathSciNet  MATH  Google Scholar 

  49. Nakamura M, Takeda Z. On the extensions of finite factors. I. Proc Japan Acad, 1959, 35: 149–154

    MathSciNet  MATH  Google Scholar 

  50. Nekrashevych V. Palindromic subshifts and simple periodic groups of intermediate growth. Ann of Math (2), 2018, 187: 667–719

    Article  MathSciNet  MATH  Google Scholar 

  51. Nekrashevych V. Simple groups of dynamical origin. Ergodic Theory Dynam Systems, 2019, 39: 707–732

    Article  MathSciNet  MATH  Google Scholar 

  52. Ol’shanskiĭ A Y. An infinite group with subgroups of prime orders. Izv Akad Nauk SSSR Ser Mat, 1980, 44: 309–321

    MathSciNet  Google Scholar 

  53. Ol’shanskiĭ A Y. Groups of bounded period with subgroups of prime order. Algebra Logic, 1982, 21: 553–618

    MathSciNet  Google Scholar 

  54. Ol’shanskiĭ A Y. On the Bass-Lubotzky question about quotients of hyperbolic groups. J Algebra, 2000, 226: 807–817

    Article  MathSciNet  MATH  Google Scholar 

  55. Ol’shanskiĭ A Y. On pairs of finitely generated subgroups in free groups. Proc Amer Math Soc, 2015, 143: 4177–4188

    Article  MathSciNet  MATH  Google Scholar 

  56. Osin D. Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems. Memoirs of the American Mathematical Society, vol. 179. Providence: Amer Math Soc, 2006

    MATH  Google Scholar 

  57. Osin D. Acylindrically hyperbolic groups. Trans Amer Math Soc, 2016, 368: 851–888

    Article  MathSciNet  MATH  Google Scholar 

  58. Ozawa N, Popa S. On a class of II1 factors with at most one Cartan subalgebra. Ann of Math (2), 2010, 172: 713–749

    Article  MathSciNet  MATH  Google Scholar 

  59. Peterson J, Thom A. Group cocycles and the ring of affiliated operators. Invent Math, 2011, 185: 561–592

    Article  MathSciNet  MATH  Google Scholar 

  60. Popa S. Maximal injective subalgebras in factors associated with free groups. Adv Math, 1983, 50: 27–48

    Article  MathSciNet  MATH  Google Scholar 

  61. Popa S. Some rigidity results for non-commutative Bernoulli shifts. J Funct Anal, 2006, 230: 273–328

    Article  MathSciNet  MATH  Google Scholar 

  62. Rădulescu F. tThe fundamental group of the von Neumann algebra of a free group with infinitely many generators is ℝ+ \ {0}. J Amer Math Soc, 1992, 5: 517–532

    MathSciNet  MATH  Google Scholar 

  63. Rădulescu F. Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index. Invent Math, 1994, 115: 347–389

    Article  MathSciNet  MATH  Google Scholar 

  64. Rips E, Segev Y, Tent K. A sharply 2-transitive group without a non-trivial abelian normal subgroup. J Eur Math Soc (JEMS), 2017, 19: 2895–2910

    Article  MathSciNet  MATH  Google Scholar 

  65. Rips E, Tent K. Sharply 2-transitive groups of characteristic 0. J Reine Angew Math, 2019, 750: 227–238

    Article  MathSciNet  MATH  Google Scholar 

  66. Rotman J J. An Introduction to the Theory of Groups. Graduate Texts in Mathematics, vol. 148. New York: Springer-Verlag, 1995

    Book  MATH  Google Scholar 

  67. Savchuk D. Some graphs related to Thompson’s group F. In: Combinatorial and Geometric Group Theory. Trends in Mathematics. Basel: Birkhäuser, 2010, 279–296

    Chapter  MATH  Google Scholar 

  68. Savchuk D. Schreier graphs of actions of Thompson’s group F on the unit interval and on the Cantor set. Geom Dedicata, 2015, 175: 355–372

    Article  MathSciNet  MATH  Google Scholar 

  69. Shen J. Maximal injective subalgebras of tensor products of free group factors. J Funct Anal, 2006, 240: 334–348

    Article  MathSciNet  MATH  Google Scholar 

  70. Sinclair A, Smith R. Finite von Neumann Algebras and Masas. London Mathematical Society Lecture Note Series, vol. 351. Cambridge: Cambridge University Press, 2008

    Book  MATH  Google Scholar 

  71. Sutherland C. Cohomology and extensions of von Neumann algebras. I. Publ Res Inst Math Sci, 1980, 16: 105–133

    Article  MathSciNet  MATH  Google Scholar 

  72. Sutherland C. Cohomology and extensions of von Neumann algebras. II. Publ Res Inst Math Sci, 1980, 16: 135–174

    Article  MathSciNet  MATH  Google Scholar 

  73. Suzuki Y. Minimal ambient nuclear C*-algebras. Adv Math, 2017, 304: 421–433

    Article  MathSciNet  MATH  Google Scholar 

  74. Suzuki Y. Non-amenable tight squeezes by Kirchberg algebras. Math Ann, 2021, in press

  75. Voiculescu D. Circular and semicircular systems and free product factors. In: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory. Progress in Mathematics, vol. 92. Boston: Birkhäuser, 1990, 45–60

    Google Scholar 

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Acknowledgements

The author was supported by the National Science Center (NCN) (Grant No. 2014/14/E/ST1/00525), Institute of Mathematics, Polish Academy of Sciences (IMPAN) from the Simons Foundation (Grant No. 346300) and the Matching 2015–2019 Polish Ministry of Science and Higher Education (MNiSW) Fund, and the Research Foundation-Flanders-Polish Academy of Sciences (FWO-PAN). The author thanks Professors Alejandra Garrido, Liming Ge, Adam Skalski, Yuhei Suzuki and Stefaan Vaes for helpful discussions. The author is also very grateful to Professors Adam Skalski and Yuhei Suzuki for reading the draft carefully, pointing out inaccuracies and providing many helpful comments.

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  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Yongle Jiang

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Jiang, Y. Maximal von Neumann subalgebras arising from maximal subgroups. Sci. China Math. 64, 2295–2312 (2021). https://doi.org/10.1007/s11425-020-1671-9

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