Polish topometric groups

Itaï Ben Yaacov (1) , Alexander Berenstein (2) , Julien Melleray (1)

1 ICJ - Institut Camille Jordan
2 UNIANDES - Departamento de Matematicas [Univ de los Andes Colombia]

We define and study the notion of \emph{ample metric generics} for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in \cite{Kechris-Rosendal:Turbulence}. Our work is based on the concept of a \emph{Polish topometric group}, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group $\Iso(\bU_1)$ of the bounded Urysohn space, the unitary group ${\mathcal U}(\ell_2)$ of a separable Hilbert space, and the automorphism group $\Aut([0,1],\lambda)$ of the Lebesgue measure algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from $\Aut([0,1],\lambda)$ into a separable topological group is continuous.

Domaines: Logique [math.LO]

Itaï Ben Yaacov, Alexander Berenstein, Julien Melleray. Polish topometric groups. Transactions of the American Mathematical Society, 2013, 365 (7), pp.3877-3897. ⟨10.1090/S0002-9947-2013-05773-X⟩. ⟨hal-00498802v3⟩