Borel Equivalence Relations
- 3.7k Downloads
This article surveys the rapidly evolving area of Borel equivalence relations under the ordering of Borel reducibility. Although the field is now considered part of descriptive set theory, it traces its origins back to areas entirely outside logic. In fact this survey starts with Silver’s theorem on the number of equivalence classes of a co-analytic equivalence relation and the landmark Harrington-Kechris-Louveau dichotomy theorem, but also takes care to sketch some of the prehistory of the subject, going back to the roots in ergodic theory, dynamics, group theory, and functional analysis.
In the later parts of the survey reader is also introduced to the theory of countable Borel equivalence relations and the connections with highly sophisticated techniques in superrigidity, the limits of structural theorems in the theory of Borel reducibility, turbulence and dynamical properties in the context of dichotomy theorems, and the connections with the theory of cardinality in the context of inner models of ZF which fail the axiom of choice.
KeywordsEquivalence Relation Dichotomy Theorem Polish Space Countable Group Countable Structure
Unable to display preview. Download preview PDF.
- Howard Becker. The number of path-components of a compact subset of ℝn. In Logic Colloquium ’95, volume 11 of Lecture Notes in Logic, pages 1–16. Springer, Berlin, 1998. Google Scholar
- Su Gao and Steve Jackson. Hyperfiniteness of abelian group actions. To appear. Google Scholar
- Su Gao and Alexander Kechris. On the classification of Polish spaces up to isometry. Memoirs of the American Mathematical Society, 161, 2003. Google Scholar
- Greg Hjorth. A dichotomy theorem for isomorphism. Preprint. Google Scholar
- Greg Hjorth. Effective cardinals. Preprint. Google Scholar
- Greg Hjorth. Group actions and countable models. In Logic Colloquium ’99, pages 3–29. Association for Symbolic Logic, Poughkeepsie, 2004. Google Scholar
- Greg Hjorth and Alexander Kechris. Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations. Memoirs of the American Mathematical Society. American Mathematical Society, Providence, 2005. Google Scholar
- Miri Segal. Hyperfiniteness. PhD thesis, Hebrew University of Jerusalem, 1997. Google Scholar
- Nicolas Shalom and Yehuda Shalom. Orbit equivalence rigidity and bounded cohomolgy. Preprint. Google Scholar