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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-2,
author = {Greg Hjorth and Alexander S. Kechris},
title = {Recent developments in the theory of Borel reducibility},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {21-52},
zbl = {0992.03055},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-2}
}
Greg Hjorth; Alexander S. Kechris. Recent developments in the theory of Borel reducibility. Fundamenta Mathematicae, Tome 167 (2001) pp. 21-52. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-2/