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 of mutually conjugate isometric actions of Γ on ℚ𝕌.
"Finitely approximable groups and actions Part II: Generic representations." J. Symbolic Logic 76 (4) 1307 - 1321, December 2011. https://doi.org/10.2178/jsl/1318338851