Isometry of Polish metric spaces

We consider the equivalence relation of isometry of separable, complete metric spaces, and show that any equivalence relation induced by a Borel action of a Polish group on a Polish space is Borel reducible to this isometry relation. We also consider the isometry relation restricted to various classes of metric spaces, and produce lower bounds for the complexity in terms of the Borel reducibility hierarchy. In this article we consider the equivalence relation of isometry of Polish metric spaces and ask how complicated it is. By a Polish metric space we mean a Polish space X together with a complete metric d on X. Two spaces are isometric if there is a bijection between them which preserves the metric. We wish to characterize the complexity of this isometry relation. We do this by considering it as an equivalence relation on an appropriately defined Polish space, and using the theory of Borel reducibility of equivalence relations. Definition 1. Let E and F be equivalence relations on the Polish spaces X and Y . We say that E is Borel reducible to Y , E ≤B Y , if there is a Borel function f : X → Y such that for all x1, x2 ∈ X we have x1 E x2 iff f(x1) F f(x2). When E is reducible to F , we may view the classification of E as at most as complicated as that of F . We will develop techniques for reducing equivalence relations induced by Borel actions of Polish groups to the isometry relation. We will show that any such equivalence relation is reducible to the isometry relation, a result obtained independently by Gao and Kechris in [8]. We will also use these techniques to find bounds on the complexity 2000 Mathematical Subject Classification. Primary 03E15; Secondary 54H05.