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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abért, Y. Glasner, and B. Virág, “Kesten’s theorem for invariant random subgroups,” Duke Math. J., 163:3 (2014), 465–488.
L. Bartholdi and R. I. Grigorchuk, “On parabolic subgroups and Hecke algebras of some fractal groups,” Serdica Math. J., 28:1 (2002), 47–90.
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.
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.
L. Bowen, “Invariant random subgroups of the free group,” Groups, Geometry, Dynamics (to appear); http://arxiv.org/abs/1204.5939.
C. Champetier, “L’espace des groupes de type fini,” Topology, 39:4 (2000), 657–680.
D. D’Angeli, A. Donno, M. Matter, and T. Nagnibeda, “Schreier graphs of the Basilica group,” J. Mod. Dyn., 4:1 (2010), 167–205.
P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 2000.
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, 134–214
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.
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.
R. I. Grigorchuk, “Construction of p-groups of intermediate growth that have a continuum of factor-groups,” Algebra i Logika, 23:4 (1984), 383–394
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.
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), 939–985
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.
R. I. Grigorchuk, “Just infinite branch groups,” in: New horizons in pro-p groups, Progr. Math., vol. 184, Birkhäuser, Boston, MA, 2000, 121–179.
R. I. Grigorchuk, “Some problems of the dynamics of group actions on rooted trees,” in: Trudy Mat. Inst. Steklov., vol. 273, Nauka, Moskva, 2011, 72–191
R. I. Grigorchuk, English transl.: in: Proc. Steklov Inst. Math., vol. 273, MAIK Nauka, Moscow, 2011, 64–175.
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.
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Math., vol. 156, Springer-Verlag, New York, 1995.
J. Lindenstrauss, G. Olsen, and Y. Sternfeld, “The Poulsen simplex,” Ann. Inst. Fourier (Grenoble), 28:1 (1978), 91–114.
A. Mann, How Groups Grow, London Math. Soc. Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012.
J. Milnor, “Problem 5603, in: Advanced Problems 5600–5609,” Amer. Math. Monthly, 75:1 (1968), 685–686.
R. Muchnik, Amenability of Universal 2-Grigorchuk group, http://arxiv.org/abs/ math/0505572.
V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence, RI, 2005.
A. Ju.šanskiĝ, “An infinite group with subgroups of prime orders,” Izv. Akad. Nauk SSSR Ser. Mat., 44:2 (1980), 309–321
A. Ju.šanskiĝ, English transl.: Math. USSR Izv., 16:2 (1981), 279–289.
D. V. Osin, “Algebraic entropy of elementary amenable groups,” Geom. Dedicata, 107 (2004), 133–151.
K. R. Parthasaraty, Probability Measures on Metric Spaces, Amer. Math. Soc. Chelsia Publishing, Providence, RI, 2005.
S. Sidki, “Automorphisms of one-rooted trees: Growth, circuit structure and acyclicity,” J. Math. Sci., 100:1 (2000), 1925–1943.
A. M. Vershik, “Nonfree actions of countable groups and their characters,” Zap. Nauchn. Sem. POMI, 378 (2010), 5–16
A. M. Vershik, English transl.: J. Math. Sci., 174:1 (2011), 1–6.
A. M. Vershik, “Totally nonfree actions and the infinite symmetric group,” Mosc. Math. J., 12:1 (2012), 193–212, 216.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
To A. M. Vershik on the occasion of his 80th birthday, with admiration and respect
The first and second authors were supported by NSF Grant DMS-1207699. The second and the third author were supported by the Swiss National Science Foundation.
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 49, No. 3, pp. 1–21, 2015
Original Russian Text Copyright © by Mustafa Gökhan Benli, Rostislav Grigorchuk, and Tatiana Nagnibeda
Rights and permissions
About this article
Cite this article
Benli, M.G., Grigorchuk, R. & Nagnibeda, T. Universal groups of intermediate growth and their invariant random subgroups. Funct Anal Its Appl 49, 159–174 (2015). https://doi.org/10.1007/s10688-015-0101-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-015-0101-4