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Recent developments in the theory of Borel reducibility

Hjorth, Greg and Kechris, Alexander S. (2001) Recent developments in the theory of Borel reducibility. Fundamenta Mathematicae, 170 (1-2). pp. 21-52. ISSN 0016-2736. https://resolver.caltech.edu/CaltechAUTHORS:20130610-142913085

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Abstract

Let E_0 be the Vitali equivalence relation and E_3 the product of countably many copies of E_0. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E_3, either E is reducible to E_0 or else E_3 is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible to a countable Borel equivalence relation or else E_3 is reducible to E. We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.


Item Type:Article
Additional Information:© 2001 Institute of Mathematics, Polish Academy of Sciences. Research of the first author partially supported by NSF Grant DMS 96-22977. Research of the second author partially supported by NSF Grant DMS 96-19880.
Funders:
Funding AgencyGrant Number
NSFDMS 96-22977
NSFDMS 96-19880
Other Numbering System:
Other Numbering System NameOther Numbering System ID
MathSciNet ReviewMR1881047
Issue or Number:1-2
Classification Code:2000 Mathematics Subject Classification: Primary 03E15
Record Number:CaltechAUTHORS:20130610-142913085
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130610-142913085
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38881
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Sep 2013 19:54
Last Modified:03 Oct 2019 05:01

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