Polish metric spaces: Their classification and isometry groups (Q2778663)

The authors announce some results on the classification of Polish spaces up to isometry. The first result says that the equivalence relation of isometry of Polish spaces is Borel bireducible with the universal relation induced by a~Borel action of a~Polish group. Further results concern some special classes of Polish spaces (i.e., spaces having some of the properties: ultrametric, \(0\)-dimensional, homogeneous, ultrahomogeneous, rigid, compact, connected, locally compact,\dots) and compute the complexity of the isometry relation for such classes. The final part of the results concerns the groups of isometries of some classes of Polish spaces. The results come from two separate sources: [\textit{J. D. Clemens}, Ph.D. thesis, University of California, Berkeley] and [\textit{S. Gao} and \textit{A. S. Kechris}, On the classification of Polish metric spaces up to isometry, preprint].