On the Classification of Polish Metric Spaces up to Isometry
colorful horizontal rule

Su Gao and Alexander S. Kechris

Abstract(preliminary)

We characterize the complexity of the classification problem of Polish metric spaces up to isometry as, up to Borel bireducibility, the unique univeral orbit equivalence relations induced by Borel actions of Polish groups. As a corollary of the proof, we derive that every Polish group is isomorphic to the isometry group of some Polish metric space.

We also develop an analysis of the isometry groups of locally compact separable metric spaces and give a complete characterization for this class of Polish groups.

Our investigation of metric structures suggests some connection of topology with model theory.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isometric Classification of Polish Metric Spaces
  4. Characterizing the Isometry Groups of Polish Metric Spaces
  5. Some Special Cases
  6. Isometries of Locally Compact Spaces, I : The Pseudo-Connected Case
  7. Isometries of Locally Compact Spaces, II : The General Case
  8. Some Analogies with the Model Theory of Countable Structures
  9. Open Problems
    References

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