Global and local boundedness of Polish groups

ABSTRACT. We present a comprehensive theory of boundedness properties for Pol-ish 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 or-bit structure of isometric or continuous affine representations on separable Banach spaces. We further study local versions of boundedness properties and the microscopic structure of Polish groups and show how the latter relates to the local dynamics of isometric and affine actions. CONTENTS 1. Global boundedness properties in Polish groups 2 1.1. Uniformities and compatible metrics 5 1.2. Constructions of linear and affine actions on Banach spaces