{"title":"(1)、(2)、(3)和(4)","authors":"Azer Akhmedov","doi":"10.1017/s0017089524000181","DOIUrl":null,"url":null,"abstract":"We prove the <jats:italic>Girth Alternative</jats:italic> for finitely generated subgroups of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000181_inline2.png\"/> <jats:tex-math> $PL_o(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also prove that a finitely generated subgroup of <jats:italic>Homeo</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000181_inline3.png\"/> <jats:tex-math> $_{+}(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> which is sufficiently rich with hyperbolic-like elements has infinite girth.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Girth Alternative for subgroups of\",\"authors\":\"Azer Akhmedov\",\"doi\":\"10.1017/s0017089524000181\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove the <jats:italic>Girth Alternative</jats:italic> for finitely generated subgroups of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000181_inline2.png\\\"/> <jats:tex-math> $PL_o(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also prove that a finitely generated subgroup of <jats:italic>Homeo</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0017089524000181_inline3.png\\\"/> <jats:tex-math> $_{+}(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> which is sufficiently rich with hyperbolic-like elements has infinite girth.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s0017089524000181\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0017089524000181","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove the Girth Alternative for finitely generated subgroups of $PL_o(I)$ . We also prove that a finitely generated subgroup of Homeo $_{+}(I)$ which is sufficiently rich with hyperbolic-like elements has infinite girth.
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