{"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}
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Abstract
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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