Skip to main content Accessibility help

Realizing uniformly recurrent subgroups

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

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.

Hide All
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed