The Theory of Countable Borel Equivalence Relations

A survey of the state of the art in a very active area of research in descriptive set theory.

Alexander S. Kechris (Author)

9781009562294, Cambridge University Press

Hardback, published 21 November 2024

176 pages
23.9 x 15.9 x 1.3 cm, 0.41 kg

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.

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.

Subject Areas: Mathematical foundations [PBC]