Polish topometric groups

TL;DR

This paper introduces the concept of ample metric generics for Polish topometric groups, explores their implications, and provides examples of groups with this property, leading to results like automatic continuity.

Contribution

It defines ample metric generics for Polish topometric groups, extending previous notions, and demonstrates their presence in key examples such as isometry, unitary, and automorphism groups.

Findings

Examples of Polish groups with ample metric generics are provided.

Ample metric generics imply the automatic continuity property for certain groups.

The concept extends the classical notion of ample generics to a metric setting.

Abstract

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.