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

Rosendal, Christian

J. Symb. Log. 76, No. 4, 1307-1321 (2011).

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)

Keywords:

Urysohn metric space; subgroup separable groups; isometry groups

Citations:

Zbl 1250.03085

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