{"title":"Torsion-free abelian groups are Borel complete | Annals of Mathematics","authors":"Gianluca Paolini, Saharon Shelah","doi":"10.4007/annals.2024.199.3.4","DOIUrl":null,"url":null,"abstract":"<p>We prove that the Borel space of torsion-free abelian groups with domain $\\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"148 1","pages":""},"PeriodicalIF":5.7000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4007/annals.2024.199.3.4","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the Borel space of torsion-free abelian groups with domain $\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.
期刊介绍：
The Annals of Mathematics is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Founded in 1884 by Ormond Stone of the University of Virginia, the journal was transferred in 1899 to Harvard University, and in 1911 to Princeton University. Since 1933, the Annals has been edited jointly by Princeton University and the Institute for Advanced Study.
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