Polish groups and Baire category methods
The article is a~modified version of the author's habilitation thesis, presenting his work on topics related to Polish groups, Baire category methods and metric model theory. Some of the results are new. In particular, it is shown that for any countably infinite group~\(G\) all conjugacy classes in the space of actions of~\(G\) on the Urysohn space are meager and that the group of bounded isometries of the Urysohn space endowed with the topology of uniform convergence is path-connected. The work consists of five sections. The first section is an introduction. In the second section, some concepts of Baire category theory and Polish groups are recalled. The third section is devoted to the Urysohn space~\(\mathbf{U}\) and the isometry group of~\(\mathbf{U}\). There is only one way of embedding~\(\mathbf{U}\) isometrically into a~normed vector space. The spaces with this rigidity property are investigated. In the fourth section, the Baire category in the space of actions of a~countable group on some homogeneous structures is studied. This concerns mainly to the space of actions of a separable Hilbert space, of the standard atomless probability algebra, and of the Urysohn space. In the last fifth section, the language of first-order logic and metric model theory play a~crucial role in the investigation of Polish groups.
scientific article; zbMATH DE number 6680206
scientific article