Glasner’s problem for Polish groups with metrizable universal minimal flow
Annales de l'Institut Fourier, to appear, 13 p.

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.

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.

Received : 2017-05-16
Revised : 2018-03-30
Accepted : 2018-04-26
Classification:  37B05,  03C1522F5054H20
Keywords: Glasner’s problem, Minimal almost periodicity, Bohr compactification
     author = {Nguyen Van Th\'e, Lionel},
     title = {Glasner's problem for Polish groups with metrizable universal minimal flow},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Nguyen Van Thé, Lionel. Glasner’s problem for Polish groups with metrizable universal minimal flow. Annales de l'Institut Fourier, to appear, 13 p.

[1] Ben Yaacov, Itaï On Roelcke precompact Polish groups which cannot act transitively on a complete metric space (2015) ( )

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

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

[4] Glasner, Shmuel Proximal flows, Springer, Lecture Notes in Mathematics, Tome 517 (1976), viii+153 pages

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

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

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

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

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

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

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

[12] Nguyen Van Thé, Lionel More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fundam. Math., Tome 222 (2013), pp. 19-47

[13] Nguyen Van Thé, Lionel On a problem of Specker about Euclidean representations of finite graphs (2017) (, to appear in Expo. Math.)

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

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

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

[17] De Vries, Jan Elements of topological dynamics, Kluwer Academic Publishers, Mathematics and its Applications, Tome 257 (1993), xvi+748 pages | Zbl 0783.54035

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

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