Global and local boundedness of Polish groups

TL;DR

This paper develops a comprehensive theory of boundedness in Polish groups, focusing on Roelcke precompactness and Property (OB), linking these properties to orbit structures and representations on metric and Banach spaces.

Contribution

It introduces a unified framework characterizing boundedness properties in Polish groups via orbit structures and representations, advancing understanding of their geometric and algebraic features.

Findings

Characterization of Roelcke precompactness through orbit structures

Linking Property (OB) to isometric and affine representations

Unified theory connecting boundedness properties with group actions

Abstract

We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on Roelcke precompactness (precompactness of the lower uniformity) and Property (OB) (boundedness of all isometric actions on separable metric spaces). In particular, these properties are characterised by the orbit structure of isometric actions on metric spaces and isometric or continuous affine representations on separable Banach spaces.