{"title":"共轭双曲子群","authors":"Nir Lazarovich, Alex Margolis, Mahan Mj","doi":"10.1090/tran/9209","DOIUrl":null,"url":null,"abstract":"<p>We show that if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fiber of a non-trivial hyperbolic bundle then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> virtually splits over a 2-ended subgroup.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":"15 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Commensurated hyperbolic subgroups\",\"authors\":\"Nir Lazarovich, Alex Margolis, Mahan Mj\",\"doi\":\"10.1090/tran/9209\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that if <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fiber of a non-trivial hyperbolic bundle then <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H\\\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> virtually splits over a 2-ended subgroup.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":\"15 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-05-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9209\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9209","RegionNum":2,"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
摘要
我们证明，如果 H H 是双曲群 G G 中一个无穷索引的非元素双曲共轭子群，那么 H H 实际上是双曲曲面群和自由群的自由积。我们证明，只要单端双曲群 H H 是非三端双曲束的纤维，那么 H H 实际上就分裂于一个二端子群之上。
We show that if HH is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group GG, then HH is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group HH is a fiber of a non-trivial hyperbolic bundle then HH virtually splits over a 2-ended subgroup.
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