Can we classify complete metric spaces up to isometry?

Abstract

We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research. Contents 1. Classification problems and Borel reducibility 2 2. The complexity of the isometry relation on separable spaces 10 3. Trees and isometry 26 4. The complexity of the isometry relation on non-separable spaces 34 5. Further extensions of the theory of Borel reducibility 41 References 42The purpose of this paper is threefold: (A) Give a concise and self-contained presentation of the theory of Borel reducibility for equivalence relations, including some motivations for its development, accessible to non-experts in the field (Section 1). (B) Illustrate how Borel reducibility (and descriptive set theory in general) may be used to gain new insight into some natural and interesting classification problems. We concentrate on the prominent example of classifying separable complete (i.e. Polish) metric spaces up to isometry, and survey the most important results in this area obtained by Gromov, Vershik, Kechris, Gao and many others, including some very recent progress on the long-standing open problem of determining the complexity of isometry between locally compact Polish metric spaces. When possible, we provide (sometimes simplified and self-contained) proofs of the most relevant results, or at least we overview the main ideas and methods involved in them (Sections 2 and 3).

Classification of One Dimensional Dynamical Systems by Countable Structures

“…The isomorphism relation of countable graphs or the isomorphism relation of countable linear orders are Borel bireducible to the S ∞ -universal orbit equivalence. The homeomorphism equivalence relation of compact metrizable spaces and the isometry relation of separable complete metric spaces were proved by Zielinski in [29] and by Melleray in [21], respectively, to be Borel bireducible to the universal orbit equivalence relation (see the survey paper by Motto Ros [23]).…”

Dynamics of metrics in measure spaces and scaling entropy

“…In § 2.3 we outline the proof of the classification theorem for metric triples, so here we provide only the precise formulation of the theorem about matrix distributions and their characterization. For the possibility of classifying metric spaces, see [34].…”