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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generic representations of countable groups

Authors: Michal Doucha and Maciej Malicki
Journal: Trans. Amer. Math. Soc. 372 (2019), 8249-8277
MSC (2010): Primary 03E15, 22F50; Secondary 20E18, 05C20
Published electronically: September 12, 2019
MathSciNet review: 4029696
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Abstract: The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $ \Gamma $ in Polish groups $ G$, i.e., elements in the Polish space $ \mathrm {Rep}(\Gamma ,G)$ of all representations of $ \Gamma $ in $ G$ whose orbits under the conjugation action of $ G$ on $ \mathrm {Rep}(\Gamma ,G)$ are comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $ K_n$-free graphs, and we show its connections with Ribes-Zalesskii-like properties of the acting groups. We prove that $ \mathbb{Z}$ has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes-Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser, and Melleray characterizing groups with a generic permutation representation.

We also investigate representations of infinite groups $ \Gamma $ in automorphism groups of metric structures such as the isometry group $ \mbox {Iso}(\mathbb{U})$ of the Urysohn space, isometry group $ \mbox {Iso}(\mathbb{U}_1)$ of the Urysohn sphere, or the linear isometry group $ \mbox {LIso}(\mathbb{G})$ of the Gurarii space. We show that the conjugation action of $ \mbox {Iso}(\mathbb{U})$ on $ \mathrm {Rep}(\Gamma ,$$ \mbox {Iso}(\mathbb{U}))$ is generically turbulent, answering a question of Kechris and Rosendal.

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Additional Information

Michal Doucha
Affiliation: Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

Maciej Malicki
Affiliation: Department of Mathematics and Mathematical Economics, Warsaw School of Economics, aleja Niepodległości 162, 02-554 Warsaw, Poland

Keywords: Group representations, generic representations, tournaments, generic turbulence, Ribes--Zalesski property
Received by editor(s): November 17, 2017
Received by editor(s) in revised form: January 17, 2018, March 8, 2019, and June 11, 2019
Published electronically: September 12, 2019
Additional Notes: The first author was supported by the GAČR project 16-34860L and RVO: 67985840.
Article copyright: © Copyright 2019 American Mathematical Society