{"title":"Embedding ℚ into a finitely presented group","authors":"James M. Belk, J. Hyde, Francesco Matucci","doi":"10.1090/bull/1762","DOIUrl":null,"url":null,"abstract":"<p>We observe that the group of all lifts of elements of Thompson’s group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\n <mml:semantics>\n <mml:mi>T</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to the real line is finitely presented and contains the additive group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the rational numbers. This gives an explicit realization of the Higman embedding theorem for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.</p>","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/bull/1762","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
We observe that the group of all lifts of elements of Thompson’s group TT to the real line is finitely presented and contains the additive group Q\mathbb {Q} of the rational numbers. This gives an explicit realization of the Higman embedding theorem for Q\mathbb {Q}, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.
期刊介绍：
The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.
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