Finitely approximable groups and actions Part II: Generic representations

2011/04/17 by Christian Rosendal, Rosendal, Christian
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO

paper · pdf · doi:10.48550/arxiv.1104.3341

arxiv created 2011/04/17 · arxiv updated 2011/04/19

Abstract

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

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