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Realizing uniformly recurrent subgroups

Published online by Cambridge University Press:  10 July 2018

NICOLÁS MATTE BON  and
TODOR TSANKOV
NICOLÁS MATTE BON
Affiliation:
D-MATH – ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland email nicolas.matte@math.ethz.ch
TODOR TSANKOV
Affiliation:
Institut de Mathématiques de Jussieu–PRG, Université Paris Diderot, 75205 Paris cedex 13, France Département de Mathématiques et Applications, École Normale Supérieure, 75005 Paris, France email todor@math.univ-paris-diderot.fr
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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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 478 - 489
Copyright
© Cambridge University Press, 2018 

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References

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), 711–790.Google Scholar
Auslander, J. and Glasner, S.. Distal and highly proximal extensions of minimal flows. Indiana Univ. Math. J. 26(4) (1977), 731–749.Google Scholar
Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465–488.Google Scholar
Berberian, S. K.. Lectures in Functional Analysis and Operator Theory (Graduate Texts in Mathematics, 15) . Springer, New York, 1974.Google Scholar
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), 67–77.Google Scholar
Chabauty, C.. Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78 (1950), 143–151.Google Scholar
Elek, G.. Uniformly recurrent subgroups and simple C ∗ -algebras. J. Funct. Anal. 274(6) (2018), 1657–1689.Google Scholar
Ellis, R.. Universal minimal sets. Proc. Amer. Math. Soc. 11 (1960), 540–543.Google Scholar
Glasner, S.. Proximal Flows (Lecture Notes in Mathematics, 517) . Springer, Berlin, 1976.Google Scholar
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. 63–75.Google Scholar
Hjorth, G.. Classification and Orbit Equivalence Relations (Mathematical Surveys and Monographs, 75) . American Mathematical Society, Providence, RI, 2000.Google Scholar
Kawabe, T.. Uniformly recurrent subgroups and the ideal structure of reduced crossed products. Preprint, 2017, arXiv:1701.03413.Google Scholar
Kennedy, M.. An intrinsic characterization of $C^{\ast }$ -simplicity. Preprint, 2015, arXiv:1509.01870v4.Google Scholar
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), 106–189.Google Scholar
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.Google Scholar
Struble, R. A.. Metrics in locally compact groups. Compos. Math. 28 (1974), 217–222.Google Scholar
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. 301–308.Google Scholar
Veech, W. A.. Topological dynamics. Bull. Amer. Math. Soc. (N.S.) 83(5) (1977), 775–830.Google Scholar