Advertisement

Functional Analysis and Its Applications

, Volume 49, Issue 3, pp 159–174 | Cite as

Universal groups of intermediate growth and their invariant random subgroups

  • Mustafa Gökhan Benli
  • Rostislav Grigorchuk
  • Tatiana Nagnibeda
Article

Abstract

We exhibit examples of groups of intermediate growth with \({2^{{\aleph _0}}}\) ergodic continuous invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.

Keywords

invariant random subgroup group of intermediate growth space of marked groups 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Abért, Y. Glasner, and B. Virág, “Kesten’s theorem for invariant random subgroups,” Duke Math. J., 163:3 (2014), 465–488.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    L. Bartholdi and R. I. Grigorchuk, “On parabolic subgroups and Hecke algebras of some fractal groups,” Serdica Math. J., 28:1 (2002), 47–90.zbMATHMathSciNetGoogle Scholar
  3. [3]
    M. G. Benli, R. Grigorchuk, and P. de la Harpe, “Amenable groups without finitely presented amenable covers,” Bull. Math. Sci., 3:1 (2013), 73–131.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    L. Bowen, R. Grigorchuk, and R. Kravchenko, “Invariant random subgroups of the lamplighter group,” Israel J. Math., 207:2 (2015), 763–782; http://arxiv.org/abs/1206.6780.MathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Bowen, “Invariant random subgroups of the free group,” Groups, Geometry, Dynamics (to appear); http://arxiv.org/abs/1204.5939.Google Scholar
  6. [6]
    C. Champetier, “L’espace des groupes de type fini,” Topology, 39:4 (2000), 657–680.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    D. D’Angeli, A. Donno, M. Matter, and T. Nagnibeda, “Schreier graphs of the Basilica group,” J. Mod. Dyn., 4:1 (2010), 167–205.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 2000.Google Scholar
  9. [9]
    R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiĭ, “Automata, dynamical systems, and groups,” in: Trudy Mat. Inst. Steklov., vol. 231, Nauka, Moskva, 2000, 134–214Google Scholar
  10. [9a]
    R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiĭ, English transl.: in: Proc. Steklov Inst. Math., vol. 231, MAIK Nauka, Moscow, 2000, 128–203.Google Scholar
  11. [10]
    R. I. Grigorchuk, “Milnor’s problem on the growth of groups and its consequences,” in: Frontiers in Complex Dynamics, Princeton Math. Ser., vol. 51, Princeton Univ. Press, Princeton, NJ, 705–773.Google Scholar
  12. [11]
    R. I. Grigorchuk, “Construction of p-groups of intermediate growth that have a continuum of factor-groups,” Algebra i Logika, 23:4 (1984), 383–394MathSciNetCrossRefGoogle Scholar
  13. [11a]
    R. I. Grigorchuk, “Construction of p-groups of intermediate growth that have a continuum of factor-groups,” English transl.: Algebra Logic, 23 (1984), 265–273.zbMATHMathSciNetGoogle Scholar
  14. [12]
    R. I. Grigorchuk, “Degrees of growth of finitely generated groups and the theory of invariant means,” Izv. Akad. Nauk SSSR Ser. Mat., 48:5 (1984), 939–985MathSciNetGoogle Scholar
  15. [12a]
    R. I. Grigorchuk, “Degrees of growth of finitely generated groups and the theory of invariant means,” English transl.: Math. USSR Izv., 25:2 (1985), 259–300.zbMATHMathSciNetGoogle Scholar
  16. [13]
    R. I. Grigorchuk, “Just infinite branch groups,” in: New horizons in pro-p groups, Progr. Math., vol. 184, Birkhäuser, Boston, MA, 2000, 121–179.MathSciNetGoogle Scholar
  17. [14]
    R. I. Grigorchuk, “Some problems of the dynamics of group actions on rooted trees,” in: Trudy Mat. Inst. Steklov., vol. 273, Nauka, Moskva, 2011, 72–191MathSciNetGoogle Scholar
  18. [14a]
    R. I. Grigorchuk, English transl.: in: Proc. Steklov Inst. Math., vol. 273, MAIK Nauka, Moscow, 2011, 64–175.zbMATHMathSciNetGoogle Scholar
  19. [15]
    R. Grigorchuk and Z. Šunić, “Self-similarity and branching in group theory,” in: Groups St. Andrews 2005, London Math. Soc. Lecture Note Series, vol. 339, Cambridge Univ. Press, Cambridge, 2007, 36–95.Google Scholar
  20. [16]
    A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Math., vol. 156, Springer-Verlag, New York, 1995.Google Scholar
  21. [17]
    J. Lindenstrauss, G. Olsen, and Y. Sternfeld, “The Poulsen simplex,” Ann. Inst. Fourier (Grenoble), 28:1 (1978), 91–114.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [18]
    A. Mann, How Groups Grow, London Math. Soc. Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012.Google Scholar
  23. [19]
    J. Milnor, “Problem 5603, in: Advanced Problems 5600–5609,” Amer. Math. Monthly, 75:1 (1968), 685–686.MathSciNetGoogle Scholar
  24. [20]
    R. Muchnik, Amenability of Universal 2-Grigorchuk group, http://arxiv.org/abs/ math/0505572.Google Scholar
  25. [21]
    V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence, RI, 2005.Google Scholar
  26. [22]
    A. Ju.šanskiĝ, “An infinite group with subgroups of prime orders,” Izv. Akad. Nauk SSSR Ser. Mat., 44:2 (1980), 309–321MathSciNetGoogle Scholar
  27. [22a]
    A. Ju.šanskiĝ, English transl.: Math. USSR Izv., 16:2 (1981), 279–289.CrossRefGoogle Scholar
  28. [23]
    D. V. Osin, “Algebraic entropy of elementary amenable groups,” Geom. Dedicata, 107 (2004), 133–151.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [24]
    K. R. Parthasaraty, Probability Measures on Metric Spaces, Amer. Math. Soc. Chelsia Publishing, Providence, RI, 2005.Google Scholar
  30. [25]
    S. Sidki, “Automorphisms of one-rooted trees: Growth, circuit structure and acyclicity,” J. Math. Sci., 100:1 (2000), 1925–1943.zbMATHMathSciNetCrossRefGoogle Scholar
  31. [26]
    A. M. Vershik, “Nonfree actions of countable groups and their characters,” Zap. Nauchn. Sem. POMI, 378 (2010), 5–16Google Scholar
  32. [26a]
    A. M. Vershik, English transl.: J. Math. Sci., 174:1 (2011), 1–6.zbMATHMathSciNetCrossRefGoogle Scholar
  33. [27]
    A. M. Vershik, “Totally nonfree actions and the infinite symmetric group,” Mosc. Math. J., 12:1 (2012), 193–212, 216.zbMATHMathSciNetGoogle Scholar
  34. [28]
    Ya. Vorobets, “Notes on the Schreier graphs of the Grigorchuk group,” in: Dynamical Systems and Group Actions, Contemporary Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012, 221–248.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Mustafa Gökhan Benli
    • 1
  • Rostislav Grigorchuk
    • 2
  • Tatiana Nagnibeda
    • 3
  1. 1.Middle East Technical UniversityAnkaraTurkey
  2. 2.Texas A&M UniversityCollege StationTexasUSA
  3. 3.Section de mathématiquesUniversité de GenèveGenèveSuisse

Personalised recommendations