## Finitely approximable groups and actions. II: Generic representations.(English)Zbl 1250.03086

Summary: Given a finitely generated group $$\Gamma$$, we study the space Isom$$(\Gamma,\mathbb {QU})$$ of all actions of $$\Gamma$$ by isometries of the rational Urysohn metric space $$\mathbb {QU}$$, where Isom$$(\Gamma,\mathbb {QU})$$ is equipped with the topology it inherits seen as a closed subset of Isom$$(\mathbb {QU})^{\Gamma }$$. When $$\Gamma$$ is the free group $$\mathbb F_{n}$$ on $$n$$ generators, this space is just Isom$$(\mathbb {QU})^{n}$$, but is in general significantly more complicated. We prove that when $$\Gamma$$ is finitely generated abelian, there is a generic point in Isom$$(\Gamma ,\mathbb {QU})$$, i.e., there is a comeagre set of mutually conjugate isometric actions of $$\Gamma$$ on $$\mathbb {QU}$$.
For Part I see [C. Rosendal, J. Symb. Log. 76, No. 4, 1297–1306 (2011; Zbl 1250.03085)].

### MSC:

 03E15 Descriptive set theory 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 54H20 Topological dynamics (MSC2010)

Zbl 1250.03085
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### References:

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