Finitely Approximate Groups and Actions Part II: Generic Representations

Abstract

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 Fn 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 ofmutually conjugate isometric actions of Γ on QU.