Israel Journal of Mathematics

, Volume 219, Issue 1, pp 215–270 | Cite as

Invariant random subgroups of linear groups



Let Γ < GL n (F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ.

Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS0(Γ) the following properties hold:

μ-almost every subgroup of Γ is open

  • F ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).

  • F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).

  • The map

Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces.

A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.

Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.

Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = <e> then Δ = <e>, μ almost surely.


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  1. [ABB+]
    M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, On the growth of l2-invariants for sequences of lattices in Lie groups, Preprint.Google Scholar
  2. [ABB+11]
    M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, On the growth of Betti numbers of locally symmetric spaces, C. R. Math. Acad. Sci. Paris 349 (2011), 831–835.MathSciNetCrossRefMATHGoogle Scholar
  3. [AGV14]
    M. Abért, Y. Glasner and B. Virág, Kesten’s theorem for invariant random subgroups, Duke Math. J. 163 (2014), 465–488.MathSciNetCrossRefMATHGoogle Scholar
  4. [AGV16]
    M. Abért, Y. Glasner and B. Virág, The measurable Kesten theorem, Ann. Probab. 44 (2016), 1601–1646.MathSciNetCrossRefMATHGoogle Scholar
  5. [AL07]
    D. Aldous and R. Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (2007), no. 54, 1454–1508.MathSciNetCrossRefMATHGoogle Scholar
  6. [BDLW16]
    U. Bader, B. Duchesne, J. Lécureux and P. Wesolek, Amenable invariant random subgroups, Israel J. Math. 213 (2016), 399–422.MathSciNetCrossRefMATHGoogle Scholar
  7. [Bek07]
    B. Bekka, Operator-algebraic superridigity for SLn(Z), n = 3, Invent. Math. 169 (2007), 401–425.MathSciNetCrossRefMATHGoogle Scholar
  8. [BG03]
    E. Breuillard and T. Gelander, On dense free subgroups of Lie groups, J. Algebra 261 (2003), 448–467.MathSciNetCrossRefMATHGoogle Scholar
  9. [BG04]
    N. Bergeron and D. Gaboriau, Asymptotique des nombres de Betti, invariants l2 et laminations, Comment. Math. Helv. 79 (2004), 362–395.MathSciNetCrossRefMATHGoogle Scholar
  10. [BG07]
    E. Breuillard and T. Gelander, A topological Tits alternative, Ann. of Math. (2) 166 (2007), 427–474.MathSciNetCrossRefMATHGoogle Scholar
  11. [BGK]
    L. Bowen, R. Grigorchuk and R. Kravchenko, Characteristic random subgroups of geometric groups and free abelian groups of infinite rank., arXiv:1402.3705, Trans. Amer. Math. Soc., to appear.Google Scholar
  12. [BGK15]
    L. Bowen, R. Grigorchuk and R. Kravchenko, Invariant random subgroups of lamplighter groups, Israel J. Math. 207 (2015), 763–782.MathSciNetCrossRefMATHGoogle Scholar
  13. [BH99]
    M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 319, Springer-Verlag, Berlin, 1999.CrossRefMATHGoogle Scholar
  14. [Bou60]
    N. Bourbaki, Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Troisième édition revue et augmentée, Actualités Sci. Indust., No. 1143. Hermann, Paris, 1960.MATHGoogle Scholar
  15. [Bow14]
    L. Bowen, Random walks on random coset spaces with applications to Furstenberg entropy, Invent. Math. 196 (2014), 485–510.MathSciNetCrossRefMATHGoogle Scholar
  16. [Bow15]
    L. Bowen, Invariant random subgroups of the free group, Groups Geom. Dyn. 9 (2015), 891–916.MathSciNetCrossRefMATHGoogle Scholar
  17. [BT72]
    F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251.CrossRefMATHGoogle Scholar
  18. [BT84]
    F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376.CrossRefMATHGoogle Scholar
  19. [Can14]
    J. Cannizzo, On invariant Schreier structures, Enseign. Math. 60 (2014), 397–415.MathSciNetCrossRefMATHGoogle Scholar
  20. [CM09]
    P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: structure theory, J. Topol. 2 (2009), 661–700.MathSciNetCrossRefMATHGoogle Scholar
  21. [CPa]
    D. Creutz and J. Peterson, Character rigidity for lattices and commensurators, arXiv:1311.4513, Trans. Amer. Math. Soc., to appear.Google Scholar
  22. [CPb]
    D. Creutz and J. Peterson, Stabilizers of ergodic actions of lattices and commensurators., arXiv:1303.3949.Google Scholar
  23. [Cre16]
    D. Creutz, Stabilizers of actions of lattices in products of groups, Ergodic Theory and Dynamical Systems electronic (2016), 1–54.Google Scholar
  24. [Fur76]
    H. Furstenberg, A note on borel’s density theorem, Proceedings of the American Mathematical Society 55 (1976), 209–212.MathSciNetMATHGoogle Scholar
  25. [GG08]
    T. Gelander and Y. Glasner, Countable primitive groups, Geom. Funct. Anal. 17 (2008), 1479–1523.MathSciNetCrossRefMATHGoogle Scholar
  26. [GM14]
    R. I. Grigorchuk and K. S. Medinets, On the algebraic properties of topological full groups, Mat. Sb. 205 (2014), 87–108.MathSciNetCrossRefGoogle Scholar
  27. [GW15]
    E. Glasner and B. Weiss, Uniformly recurrent subgroups, in Recent trends in ergodic theory and dynamical systems, Contemp. Math., Vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 63–75.Google Scholar
  28. [HT16]
    Y. Hartman and O. Tamuz, Stabilizer rigidity in irreducible group actions, Israel J. Math. 216 (2016), 679–705.MathSciNetCrossRefMATHGoogle Scholar
  29. [Kec95]
    A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
  30. [KN13]
    I. Kapovich and T. Nagnibeda, Subset currents on free groups, Geom. Dedicata 166 (2013), 307–348.MathSciNetCrossRefMATHGoogle Scholar
  31. [Lan02]
    S. Lang, Algebra, third ed., Graduate Texts in Mathematics, Vol. 211, Springer- Verlag, New York, 2002.CrossRefMATHGoogle Scholar
  32. [LP15]
    R. Lyons and Y. Peres, Cycle density in infinite Ramanujan graphs, Ann. Probab. 43 (2015), 3337–3358.MathSciNetCrossRefMATHGoogle Scholar
  33. [MS79]
    G. A. Margulis and G. A. Soĭfer, Nonfree maximal subgroups of infinite index of the group SLn(Z), Uspekhi Mat. Nauk 34 (1979), 203–204.MathSciNetGoogle Scholar
  34. [MS81]
    G. A. Margulis and G. A. Soĭfer, Maximal subgroups of infinite index in finitely generated linear groups, J. Algebra 69 (1981), 1–23.MathSciNetCrossRefMATHGoogle Scholar
  35. [Par00]
    A. Parreau, Dégénérescences de sous-groupes discrets de groupes de lie semisimples et actions de groupes sur des immeubles affines,, Ph.D. thesis, University Paris 11, Orsay,, 2000.Google Scholar
  36. [PT16]
    J. Peterson and A. Thom, Character rigidity for special linear groups, J. Reine Angew. Math. 716 (2016), 207–228.MathSciNetMATHGoogle Scholar
  37. [Rai]
    J. Raimbault, On the convergence of arithmetic orbifolds., arXiv:1311.5375.Google Scholar
  38. [RTW15]
    B. Rémy, A. Thuillier and A. Werner, Bruhat-Tits buildings and analytic geometry, in Berkovich spaces and applications, Lecture Notes in Math., Vol. 2119, Springer, Cham, 2015, pp. 141–202.Google Scholar
  39. [Sha99]
    Y. Shalom, Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T), Trans. Amer. Math. Soc. 351 (1999), 3387–3412.CrossRefMATHGoogle Scholar
  40. [SZ94]
    G. Stuck and R. J. Zimmer, Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2) 139 (1994), 723–747.MathSciNetCrossRefMATHGoogle Scholar
  41. [TD]
    R. Tucker-Drob, Shift-minimal groups, fixed price 1, and the unique trace property, arXiv:1211.6395.Google Scholar
  42. [Tit72]
    J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.MathSciNetCrossRefMATHGoogle Scholar
  43. [TTD14]
    S. Thomas and R. Tucker-Drob, Invariant random subgroups of strictly diagonal limits of finite symmetric groups, Bull. Lond. Math. Soc. 46 (2014), 1007–1020.MathSciNetCrossRefMATHGoogle Scholar
  44. [Ver10]
    A. M. Vershik, Nonfree actions of countable groups and their characters, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 378 (2010), 5–16, 228.Google Scholar
  45. [Ver12]
    A. M. Vershik, Totally nonfree actions and the infinite symmetric group, Mosc. Math. J. 12 (2012), 193–212, 216.MathSciNetMATHGoogle Scholar
  46. [Weh73]
    B. A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Springer-Verlag, New York, 1973, Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76.MATHGoogle Scholar
  47. [Wei95]
    A. Weil, Basic number theory, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the second (1973) edition.Google Scholar

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael

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