Generic representations of countable groups

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Generic representations of countable groups

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Generic representations of countable groups

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF16-34860L" target="_blank" >GF16-34860L: Logika a topologie v Banachových prostorech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

—

Svazek periodika

372

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

28

Strana od-do

8249-8277

Kód UT WoS článku

000514303900025

EID výsledku v databázi Scopus

2-s2.0-85075129194