Finitely approximable groups and actions Part II: Generic representations

الأصل · EN

Given a finitely generated group Γ, we study the space Isom(Γ,QU) of all actions of Γ by isometries of the rational Urysohn metric space QU, where Isom(Γ,QU) is equipped with the topology it inherits seen as a closed subset of Isom(QU)Γ. When Γ is the free group ₙ on n generators this space is just Isom(QU)ⁿ, but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ,QU), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on QU.