The theory of definable equivalence relations has been a vibrant area of research in descriptive set theory for the past three decades. It serves as a foundation of a theory of complexity of classification problems in mathematics and is further motivated by the study of group actions in a descriptive, topological, or measure-theoretic context. A key part of this theory is concerned with the structure of countable Borel equivalence relations. These are exactly the equivalence relations generated by Borel actions of countable discrete groups and this introduces important connections with group theory, dynamical systems, and operator algebras. This text surveys the state of the art in the theory of countable Borel equivalence relations and delineates its future directions and challenges. It gives beginning graduate students and researchers a bird's-eye view of the subject, with detailed references to the extensive literature provided for further study.
This book surveys the state of the art in a very active research area in descriptive set theory, which has important connections to many areas of mathematics, including group theory, dynamical systems, and operator algebras. It will be of great value to beginning graduate students and researchers in these areas.
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A survey of the state of the art in a very active area of research in descriptive set theory.
1. Equivalence relations and reductions;
2. Countable Borel equivalence relations;
3. Essentially countable relations;
4. Invariant and quasi-invariant measures;
5. Smoothness, $\mathbf{E}_0$ and $\mathbf{E}_\infty$;
6. Rigidity and incomparability;
7. Hyperfiniteness;
8. Amenability;
9. Treeability;
10. Freeness;
11. Universality;
12. The poset of bireducibility types;
13. Structurability;
14. Topological realizations;
15. A universal space for actions and equivalence relations;
16. Open problems; References; List of Notation; Subject Index.
Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the Sloan Research Fellowship, the J. S. Guggenheim Memorial Foundation Fellowship, and the Carol Karp Prize of the Association for Symbolic Logic. He is also an Inaugural Fellow of the American Mathematical Society.
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