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Finitely approximable groups and actions Part II: Generic representations
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- Journal:
- The Journal of Symbolic Logic / Volume 76 / Issue 4 / December 2011
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1307-1321
- Print publication:
- December 2011
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- Article
- Export citation
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Given a finitely generated group Γ we study the space Isom(Γ,
) of all actions of Γ by isometries of the rational Urysohn metric space 
, where Isom (Γ, 
) is equipped with the topology it inherits seen as a closed subset of Isom 
. When Γ is the free group 
on n generators this space is just Isom 
, but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ, 
), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on 
.
) of all actions of Γ by isometries of the rational Urysohn metric space 
, where Isom (Γ, 
) is equipped with the topology it inherits seen as a closed subset of Isom 
. When Γ is the free group 
on 
, but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ, 
), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on 
.