Generic representations of countable groups

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00512066" target="_blank" >RIV/67985840:_____/19:00512066 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1090/tran/7932" target="_blank" >http://dx.doi.org/10.1090/tran/7932</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/tran/7932" target="_blank" >10.1090/tran/7932</a>

Result language
angličtina
Original language name
Generic representations of countable groups
Original language description
The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups Γ in Polish groups G, i.e., elements in the Polish space Rep(Γ, G) of all representations of Γ in G whose orbits under the conjugation action of G on Rep(Γ, G) are comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or Kn-free graphs, and we show its connections with Ribes-Zalesskii-like properties of the acting groups. We prove that 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 Γ in automorphism groups of metric structures such as the isometry group Iso(U) of the Urysohn space, isometry group Iso(U1) of the Urysohn sphere, or the linear isometry group LIso(G) of the Gurarii space. We show that the conjugation action of Iso(U) on Rep(Γ, Iso(U)) is generically turbulent, answering a question of Kechris and Rosendal.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF16-34860L" target="_blank" >GF16-34860L: Logic and Topology in Banach spaces</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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