§ 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X, d).Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task.After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001].

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We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We prove that for every Borel equivalence relation E, either E is Borel reducible to E 0 , or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E, either E ∈ F or F is Borel reducible to E, then F consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.2010 Mathematics Subject Classification. Primary 03E15; secondary 28A05.

We use the theory of Borel equivalence relations to analyze the equivalence relation of isomorphism among one-dimensional subshifts. We show that this equivalence relation is a universal countable Borel equivalence relation, so that it admits no definable complete invariants fundamentally simpler than the equivalence classes. We also see that the classification of higher dimensional subshifts up to isomorphism has the same complexity as for the one-dimensional case.The problem of classifying one-dimensional and higher dimensional subshifts has been well-studied, with the aim of finding invariants for isomorphism. One can consider this equivalence relation from the standpoint of descriptive set theory, and consider its complexity among Borel equivalence relations under the relation of Borel reducibility, ≤ B .Definition 1: Let A be a finite set of symbols. A one-dimensional subshift on A is a closed subset of A Z which is invariant under the shift operator S, where S(x)(n) = x(n + 1). Two subshifts X on A and Y on B are isomorphic if there is a homeomorphism ϕ : X → Y which commutes with S.A subshift may be defined by a set of forbidden words W ⊆ A