Finitely Approximate Groups and Actions Part II: Generic Representations
journal contributionposted on 2012-08-14, 00:00 authored by Christian Rosendal
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 ofmutually conjugate isometric actions of Γ on QU.
The author was partially supported by NSF grants DMS 0901405 and DMS 0919700. The author is also grateful for the many helpful suggestions of the referee.
Publisher Statement© 2011, Association for Symbolic Logic. The original version is available through Association for Symbolic Logic at DOI: 644mpo9untexw1`10.2178/jsl/1318338851
PublisherAssociation for Symbolic Logic