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Realizing uniformly recurrent subgroups

  • NICOLÁS MATTE BON (a1) and TODOR TSANKOV (a2) (a3)
Abstract

We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notion of a universal minimal flow relative to a uniformly recurrent subgroup and prove its existence and uniqueness.

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[ABB+] Abért, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J. and Samet, I.. On the growth of L 2 -invariants for sequences of lattices in Lie groups. Ann. of Math. (2) 185(3) (2017), 711790.
[AG] Auslander, J. and Glasner, S.. Distal and highly proximal extensions of minimal flows. Indiana Univ. Math. J. 26(4) (1977), 731749.
[AGV] Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465488.
[B] Berberian, S. K.. Lectures in Functional Analysis and Operator Theory (Graduate Texts in Mathematics, 15) . Springer, New York, 1974.
[BYMT] Ben Yaacov, I., Melleray, J. and Tsankov, T.. Metrizable universal minimal flows of Polish groups have a comeagre orbit. Geom. Funct. Anal. 27(1) (2017), 6777.
[C] Chabauty, C.. Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78 (1950), 143151.
[E1] Elek, G.. Uniformly recurrent subgroups and simple C -algebras. J. Funct. Anal. 274(6) (2018), 16571689.
[E2] Ellis, R.. Universal minimal sets. Proc. Amer. Math. Soc. 11 (1960), 540543.
[G] Glasner, S.. Proximal Flows (Lecture Notes in Mathematics, 517) . Springer, Berlin, 1976.
[GW] Glasner, E. and Weiss, B.. Uniformly recurrent subgroups. Recent Trends in Ergodic Theory and Dynamical Systems (Contemporary Mathematics, 631) . American Mathematical Society, Providence, RI, 2015, pp. 6375.
[H] Hjorth, G.. Classification and Orbit Equivalence Relations (Mathematical Surveys and Monographs, 75) . American Mathematical Society, Providence, RI, 2000.
[K1] Kawabe, T.. Uniformly recurrent subgroups and the ideal structure of reduced crossed products. Preprint, 2017, arXiv:1701.03413.
[K2] Kennedy, M.. An intrinsic characterization of $C^{\ast }$ -simplicity. Preprint, 2015, arXiv:1509.01870v4.
[KPT] Kechris, A. S., Pestov, V. G. and Todorčević, S.. Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15(1) (2005), 106189.
[LBMB] Le Boudec, A. and Matte Bon, N.. Subgroup dynamics and C -simplicity of groups of homeomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 51(3) (2018), in press.
[S] Struble, R. A.. Metrics in locally compact groups. Compos. Math. 28 (1974), 217222.
[U] Uspenskij, V.. On universal minimal compact G-spaces. Proc. 2000 Topology and Dynamics Conf. (San Antonio, TX). Vol. 25. Auburn University, Auburn, AL, 2000, pp. 301308.
[V] Veech, W. A.. Topological dynamics. Bull. Amer. Math. Soc. (N.S.) 83(5) (1977), 775830.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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