Isometry of Polish metric spaces (Q435193)

The author shows that any equivalence relation induced by a Polish group action can be Borel reduced to the relation of isometry of Polish metric spaces. This result has been independently obtained by \textit{S. Gao} and \textit{A. S. Kechris} [``On the classification of Polish metric spaces up to isometry'', Mem. Am. Math. Soc. 766, 78 p. (2003; Zbl 1012.54038)], who also showed that the isometry of Polish metric spaces is Borel reducible to an action of a Polish group. These results imply that the relation of isometry of Polish metric spaces is a universal orbit equivalene relation induced by a Polish group action.NEWLINENEWLINEThe techniques of this paper give a direct construction of a Polish metric space from a continuous action of a Polish group. This method is then applied to obtain lower bounds on the complexity of the isometry relation restricted to certain classes of metric spaces. The author shows that isometry of Polish zero-dimensional spaces is not classifiable by countable structures, which contrasts a result of Gao and Kechris saying that the isometry of Polish zero-dimensional locally compact metric spaces is bireducible with graph isomorphism. Similarly, the author shows that isometry of ultra-homogeneous Polish metric spaces is not classifiable by countable structures and that isometry of homogeneous discrete Polish metric spaces is not smooth.