{"title":"流形的简单体积,其基本群在无穷远处是可调和的","authors":"Giuseppe Bargagnati","doi":"10.1017/s0017089524000107","DOIUrl":null,"url":null,"abstract":"We show that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline1.png\" /> <jats:tex-math> $n \\neq 1,4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the simplicial volume of an inward tame triangulable open <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline2.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-manifold <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline3.png\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with amenable fundamental group at infinity at each end is finite; moreover, we show that if also <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline4.png\" /> <jats:tex-math> $\\pi _1(M)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is amenable, then the simplicial volume of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline5.png\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"27 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Simplicial volume of manifolds with amenable fundamental group at infinity\",\"authors\":\"Giuseppe Bargagnati\",\"doi\":\"10.1017/s0017089524000107\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000107_inline1.png\\\" /> <jats:tex-math> $n \\\\neq 1,4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the simplicial volume of an inward tame triangulable open <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000107_inline2.png\\\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-manifold <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000107_inline3.png\\\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with amenable fundamental group at infinity at each end is finite; moreover, we show that if also <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000107_inline4.png\\\" /> <jats:tex-math> $\\\\pi _1(M)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is amenable, then the simplicial volume of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000107_inline5.png\\\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.\",\"PeriodicalId\":50417,\"journal\":{\"name\":\"Glasgow Mathematical Journal\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-04-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Glasgow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s0017089524000107\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0017089524000107","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Simplicial volume of manifolds with amenable fundamental group at infinity
We show that for $n \neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
The journal has a web-based submission system for articles. For details of how to to upload your paper see GMJ - Online Submission Guidelines or go directly to the submission site.