{"title":"关于群图的有限性质的注记","authors":"Frédéric Haglund, D. Wise","doi":"10.1090/BPROC/81","DOIUrl":null,"url":null,"abstract":"<p>We show that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}_n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> splits as a finite graph of groups, then the vertex groups are of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}_n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> if the edge groups are of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F Subscript n\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {F}_n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on finiteness properties of graphs of groups\",\"authors\":\"Frédéric Haglund, D. Wise\",\"doi\":\"10.1090/BPROC/81\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is of type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F Subscript n\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}_n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> splits as a finite graph of groups, then the vertex groups are of type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F Subscript n\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}_n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> if the edge groups are of type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper F Subscript n\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {F}_n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"17 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-03-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/BPROC/81\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/BPROC/81","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了如果G G是F n \mathcal {F}_n型，并且G G分裂为群的有限图，那么如果边群是F n \mathcal {F}_n型，则顶点群是F n \mathcal {F}_n型。
A note on finiteness properties of graphs of groups
We show that if GG is of type Fn\mathcal {F}_n, and GG splits as a finite graph of groups, then the vertex groups are of type Fn\mathcal {F}_n if the edge groups are of type Fn\mathcal {F}_n.
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