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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Generic representations of countable groups
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by Michal Doucha and Maciej Malicki PDF
Trans. Amer. Math. Soc. 372 (2019), 8249-8277 Request permission

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
  • MR Author ID: 984180
  • Email: doucha@math.cas.cz
  • Maciej Malicki
  • Affiliation: Department of Mathematics and Mathematical Economics, Warsaw School of Economics, aleja Niepodległości 162, 02-554 Warsaw, Poland
  • MR Author ID: 756387
  • Email: mamalicki@gmail.com
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8249-8277
  • MSC (2010): Primary 03E15, 22F50; Secondary 20E18, 05C20
  • DOI: https://doi.org/10.1090/tran/7932
  • MathSciNet review: 4029696