2017/10/23 by Doucha, Michal, Malicki, Maciej · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)
paper · doi:10.48550/arxiv.1710.08170
The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups Γ in Polish groups G, i.e. those elements in the Polish space Rep(Γ,G) of all representations of Γ in G, whose orbit under the conjugation action of G on Rep(Γ,G) is 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-Zalesski-like properties of the acting groups. We prove that ℕ 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(\mathbbU) of the Urysohn space, isometry group Iso(\mathbbU1) of the Urysohn sphere, or the linear isometry group LIso(\mathbbG) of the Gurarii space. We show that the conjugation action of Iso(\mathbbU) on Rep(Γ,Iso(\mathbbU)) is generically turbulent answering a question of Kechris and Rosendal.