Glasner’s problem for Polish groups with metrizable universal minimal flow
[Le problème de Glasner pour les groupes polonais dont le flot minimal universel est métrisable]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 941-953.

Un problème dû à Glasner, et désormais connu sous de nom de problème de Glasner, demande s’il existe un groupe polonais, minimalement presque périodique et monothétique, qui n’est pas extrêmement moyennable. Le but de cette courte note est d’observer qu’une réponse négative s’obtient sous l’hypothèse supplémentaire de la métrisablité du flot minimal universel.

Avertissement :

Suite à une erreur d’édition, la référence [13] est erronée dans le PDF. Elle aurait dû être :

[13] Nguyen Van Thé, Lionel On a problem of Specker about Euclidean representations of finite graphs (2017) (https://arxiv.org/abs/0810.2359, to appear in Expo. Math.)

A problem of Glasner, now known as Glasner’s problem, asks whether there exists a minimally almost periodic, monothetic, Polish group that is not extremely amenable. The purpose of this short note is to observe that a negative answer is obtained under the additional assumption that the universal minimal flow is metrizable.

Disclaimer:

Due to an editorial error, the citation [13] is wrong in the PDF. It should read:

[13] Nguyen Van Thé, Lionel On a problem of Specker about Euclidean representations of finite graphs (2017) (https://arxiv.org/abs/0810.2359, to appear in Expo. Math.)

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DOI : https://doi.org/10.5802/aif.3262
Classification : 37B05,  03C1522F5054H20
Mots clés : Problème de Glasner, presque périodicité minimale, compactification de Bohr
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     author = {Nguyen Van Th\'e, Lionel},
     title = {Glasner{\textquoteright}s problem for Polish groups with metrizable universal minimal flow},
     journal = {Annales de l'Institut Fourier},
     pages = {941--953},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {2},
     year = {2019},
     doi = {10.5802/aif.3262},
     zbl = {07067423},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3262/}
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Nguyen Van Thé, Lionel. Glasner’s problem for Polish groups with metrizable universal minimal flow. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 941-953. doi : 10.5802/aif.3262. http://archive.numdam.org/articles/10.5802/aif.3262/

[1] Ben Yaacov, Itaï On Roelcke precompact Polish groups which cannot act transitively on a complete metric space (2015) (https://hal.archives-ouvertes.fr/hal-01207953v2) | Zbl 1400.37030

[2] Ben Yaacov, Itaï; Melleray, Julien; Tsankov, Todor Metrizable universal minimal flows of Polish groups have a comeagre orbit, Geom. Funct. Anal., Volume 27 (2017) no. 1, pp. 67-77 | MR 3613453 | Zbl 1364.54026

[3] Glasner, Eli On minimal actions of Polish groups, Topology Appl., Volume 85 (1998) no. 1-3, pp. 119-125 | Article | MR 1617456 | Zbl 0923.54030

[4] Glasner, Shmuel Proximal flows, Lecture Notes in Mathematics, 517, Springer, 1976, viii+153 pages | MR 474243 | Zbl 0322.54017

[5] Kazhdan, David On the connection of the dual space of a group with the structure of its closed subgroups, Funkts. Anal. Prilozh., Volume 1 (1967), pp. 71-74 | MR 209390 | Zbl 0168.27602

[6] Kechris, Alexander S.; Pestov, Vladimir G.; Todorcevic, Stevo Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 106-189 | Zbl 1084.54014

[7] Lachlan, Alistair H. Countable homogeneous tournaments, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 431-461 | Article | MR 743728 | Zbl 0562.05025

[8] Lachlan, Alistair H.; Woodrow, Robert E. Countable ultrahomogeneous undirected graphs, Trans. Am. Math. Soc., Volume 262 (1980) no. 1, pp. 51-94 | Article | MR 583847 | Zbl 0471.03025

[9] Macpherson, Dugald; Tent, Katrin Simplicity of some automorphism groups, J. Algebra, Volume 342 (2011) no. 1, pp. 40-52 | MR 2824528 | Zbl 1244.20002

[10] Melleray, Julien; Nguyen Van Thé, Lionel; Tsankov, Todor Polish groups with metrizable universal minimal flows, Int. Math. Res. Not., Volume 2016 (2016) no. 5, pp. 1285-1307 | Article | MR 3509926 | Zbl 1359.37023

[11] Nešetřil, Jaroslav; Rödl, Vojtěch Partitions of finite relational and set systems, J. Comb. Theory, Ser. A, Volume 22 (1977) no. 3, pp. 289-312 | MR 437351 | Zbl 0361.05017

[12] Nguyen Van Thé, Lionel More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fundam. Math., Volume 222 (2013), pp. 19-47 | Article | MR 3080786 | Zbl 1293.37006

[13] Nguyen Van Thé, Lionel On a problem of Specker about Euclidean representations of finite graphs (2017) (https://arxiv.org/abs/0810.2359, to appear in Expo. Math.)

[14] Open problems in topology. II (Pearl, Elliott, ed.), Elsevier, 2007, xii+763 pages | MR 2367385 | Zbl 1158.54300

[15] Truss, John K. The group of the countable universal graph, Math. Proc. Camb. Philos. Soc., Volume 98 (1985) no. 2, pp. 213-245 | MR 795890 | Zbl 0586.20004

[16] Tsankov, Todor Unitary representations of oligomorphic groups, Geom. Funct. Anal., Volume 22 (2012) no. 2, pp. 528-555 | Article | MR 2929072 | Zbl 1252.22003

[17] de Vries, Jan Elements of topological dynamics, Mathematics and its Applications, 257, Kluwer Academic Publishers, 1993, xvi+748 pages | MR 1249063 | Zbl 0783.54035

[18] Wang, Paul S. On isolated points in the dual spaces of locally compact groups, Math. Ann., Volume 218 (1975) no. 1, pp. 19-34 | MR 384993 | Zbl 0332.22009

[19] Zucker, Andy Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem, Trans. Am. Math. Soc., Volume 368 (2016) no. 9, pp. 6715-6740 | Article | MR 3461049 | Zbl 1359.37024