{"title":"将π嵌入到有限表示群中","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":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"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\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2022-08-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/bull/1762\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/bull/1762","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 8
摘要
我们观察到Thompson群T的元素到实数的所有提升的群是有限的，并且包含有理数的加法群Q\mathbb｛Q｝。这给出了Q\mathbb｛Q｝的Higman嵌入定理的显式实现，回答了Martin Bridson和Pierre de la Harpe的Kourovka笔记本问题。
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.
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