The Theory of Countable Borel Equivalence Relations

Alexander S. Kechris (California Institute of Technology)

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English

Cambridge University Press
21 November 2024

Set theory; Groups & group theory; Functional analysis & transforms

Series: Cambridge Tracts in Mathematics

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.

By: Alexander S. Kechris (California Institute of Technology)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 229mm, Width: 152mm, Spine: 11mm
Weight: 403g
ISBN: 9781009562294
ISBN 10: 1009562290
Series: Cambridge Tracts in Mathematics
Pages: 174
Publication Date: 21 November 2024
Audience: College/higher education , Further / Higher Education
Format: Hardback
Publisher's Status: Active

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; 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.