{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:08:29Z","timestamp":1649203709131},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3114,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2005,9]]},"abstract":"This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:<\/jats:p>Theorem<\/jats:sc> 0.1. There is an essentially countable Borel equivalence relation E such that for<\/jats:italic> no countable Borel equivalence relation F<\/jats:italic> (on a standard Borel space<\/jats:italic>) do we have<\/jats:italic><\/jats:p><\/jats:disp-formula><\/jats:p>The proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the following<\/jats:p>Messy<\/jats:sc> Fact<\/jats:sc> 0.2. There are countable Borel equivalence relations<\/jats:italic>such that<\/jats:italic>:<\/jats:p>(i) each<\/jats:italic>Ex<\/jats:sub><\/jats:italic>is defined on a standard Borel probability space (Xx<\/jats:sub><\/jats:italic>, \u03bcx<\/jats:sub><\/jats:italic>); each Ex<\/jats:sub><\/jats:italic> is \u03bcx<\/jats:sub><\/jats:italic>-invariant and \u03bcx<\/jats:sub><\/jats:italic>-ergodic;<\/jats:italic><\/jats:p>(ii) for<\/jats:italic>x<\/jats:italic>1<\/jats:sub> \u2260 x<\/jats:italic>2<\/jats:sub> and A<\/jats:italic> \u03bcx<\/jats:italic>\u03b9<\/jats:sub> -conull, we have<\/jats:italic>E<\/jats:italic>x<\/jats:italic>\u03b9<\/jats:sub>\/A<\/jats:italic><\/jats:sub>not Borel reducible to<\/jats:italic>E<\/jats:italic>x<\/jats:italic>2<\/jats:sub>;<\/jats:p>(iii) if f<\/jats:italic>: Xx<\/jats:sub><\/jats:italic> \u2192 Xx<\/jats:sub><\/jats:italic>is a measurable reduction of<\/jats:italic>Ex<\/jats:sub><\/jats:italic>to itself then<\/jats:italic><\/jats:p>(iv)<\/jats:p><\/jats:disp-formula><\/jats:p>is a standard Borel space on which the projection function<\/jats:italic><\/jats:p><\/jats:disp-formula><\/jats:p>is Borel and the equivalence relation \u00ca given by<\/jats:italic><\/jats:p><\/jats:disp-formula><\/jats:p>if and only if<\/jats:italic>x<\/jats:italic> = x<\/jats:italic>\u2032 and<\/jats:italic>zEx<\/jats:sub>z<\/jats:italic>\u2032 is Borel<\/jats:italic>;<\/jats:p>(V)<\/jats:p><\/jats:disp-formula><\/jats:p>is Borel<\/jats:italic>.<\/jats:p>We first prove the theorem granted this messy fact. We then prove the fact.<\/jats:p>(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).<\/jats:p>Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.<\/jats:p>","DOI":"10.2178\/jsl\/1122038924","type":"journal-article","created":{"date-parts":[[2005,7,22]],"date-time":"2005-07-22T18:48:29Z","timestamp":1122058109000},"page":"979-992","source":"Crossref","is-referenced-by-count":1,"title":["Bi-Borel reducibility of essentially countable Borel equivalence relations"],"prefix":"10.1017","volume":"70","author":[{"given":"Greg","family":"Hjorth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006897_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(02)00068-4"},{"key":"S0022481200006897_ref004","unstructured":"Monod N. and Shalom Y. , Orbit equivalence and bounded cohomology, to appear in the Annals of Mathematics , available at http:\/\/www.math.uchicago.edu\/~monod\/publications.html."},{"key":"S0022481200006897_ref003","volume-title":"Classical descriptive set theory","volume":"156","author":"Kechris","year":"1994"},{"key":"S0022481200006897_ref002","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061302000138"},{"key":"S0022481200006897_ref001","unstructured":"Hjorth G. and Kechris A. S. , Rigidity theorems for actions of product groups and countable Borel equivalence relations, to appear in the Memoirs of the American Mathematical Society , available at http:\/\/www.math.caltech.edu\/people\/kechris.html."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200006897","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,3]],"date-time":"2019-05-03T20:14:53Z","timestamp":1556914493000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200006897\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9]]},"references-count":5,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2005,9]]}},"alternative-id":["S0022481200006897"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1122038924","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,9]]}}}