Generic representations of countable groups

0512066 - MÚ 2020 RIV US eng J - Článek v odborném periodiku
Doucha, Michal - Malicki, M.
Generic representations of countable groups.
American Mathematical Society. Transactions. Roč. 372, č. 11 (2019), s. 8249-8277. ISSN 0002-9947. E-ISSN 1088-6850
Grant CEP: GA ČR GF16-34860L
Institucionální podpora: RVO:67985840
Klíčová slova: group representations * generic representations * tournaments * generic turbulence * Ribes--Zalesski property
Obor OECD: Pure mathematics
Impakt faktor: 1.363, rok: 2019
Způsob publikování: Omezený přístup
http://dx.doi.org/10.1090/tran/7932

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.
Trvalý link: http://hdl.handle.net/11104/0302269