# Recent developments in the theory of Borel reducibility

Greg Hjorth; Alexander S. Kechris

Fundamenta Mathematicae (2001)

- Volume: 170, Issue: 1-2, page 21-52
- ISSN: 0016-2736

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topGreg Hjorth, and Alexander S. Kechris. "Recent developments in the theory of Borel reducibility." Fundamenta Mathematicae 170.1-2 (2001): 21-52. <http://eudml.org/doc/281714>.

@article{GregHjorth2001,

abstract = {
Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ 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 E₃ is reducible to E.
We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.
},

author = {Greg Hjorth, Alexander S. Kechris},

journal = {Fundamenta Mathematicae},

keywords = {Borel equivalence relation; Borel reducibility; dichotomy theorems},

language = {eng},

number = {1-2},

pages = {21-52},

title = {Recent developments in the theory of Borel reducibility},

url = {http://eudml.org/doc/281714},

volume = {170},

year = {2001},

}

TY - JOUR

AU - Greg Hjorth

AU - Alexander S. Kechris

TI - Recent developments in the theory of Borel reducibility

JO - Fundamenta Mathematicae

PY - 2001

VL - 170

IS - 1-2

SP - 21

EP - 52

AB -
Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ 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 E₃ is reducible to E.
We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.

LA - eng

KW - Borel equivalence relation; Borel reducibility; dichotomy theorems

UR - http://eudml.org/doc/281714

ER -

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