Global and local boundedness of Polish groups Christian Rosendal Primary: 22A25Secondary: 03E1546B04Boundedness of topological groupsaffine actions on Banach spaces We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on \emph{Roelcke precompactness} (precompactness of the lower uniformity) and \emph{Property (OB)} (boundedness of all isometric actions on separable metric spaces). In particular, these properties are characterised by the orbit 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5133 10.1512/iumj.2013.62.5133 en Indiana Univ. Math. J. 62 (2013) 1621 - 1678 state-of-the-art mathematics http://iumj.org/access/