## 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