Finitely approximable 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 𝔽_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.