{"title":"A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC","authors":"LONGYUN DING, XU WANG","doi":"10.1017/jsl.2024.7","DOIUrl":null,"url":null,"abstract":"<p>In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span>-CLI and L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span>-CLI where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span> is a countable ordinal. We establish three results: </p><ol><li><p><span>(1)</span> <span>G</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span>-CLI iff <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$G=\\{1_G\\}$</span></span></img></span></span>;</p></li><li><p><span>(2)</span> <span>G</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span>-CLI iff <span>G</span> admits a compatible complete two-sided invariant metric; and</p></li><li><p><span>(3)</span> <span>G</span> is L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span>-CLI iff <span>G</span> is locally <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span>-CLI, i.e., <span>G</span> contains an open subgroup that is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></img></span></span>-CLI.</p></li></ol><p></p><p>Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$G_\\alpha $</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$H_\\alpha $</span></span></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline12.png\"/><span data-mathjax-type=\"texmath\"><span>$\\alpha <\\omega _1$</span></span></span></span>, such that: </p><ol><li><p><span>(1)</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline13.png\"/><span data-mathjax-type=\"texmath\"><span>$H_\\alpha $</span></span></span></span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline14.png\"/><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></span></span>-CLI but not L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline15.png\"/><span data-mathjax-type=\"texmath\"><span>$\\beta $</span></span></span></span>-CLI for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline16.png\"/><span data-mathjax-type=\"texmath\"><span>$\\beta <\\alpha $</span></span></span></span>; and</p></li><li><p><span>(2)</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline17.png\"/><span data-mathjax-type=\"texmath\"><span>$G_\\alpha $</span></span></span></span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline18.png\"/><span data-mathjax-type=\"texmath\"><span>$(\\alpha +1)$</span></span></span></span>-CLI but not L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline19.png\"/><span data-mathjax-type=\"texmath\"><span>$\\alpha $</span></span></span></span>-CLI.</p></li></ol><p></p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":"27 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/jsl.2024.7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $-CLI and L-$\alpha $-CLI where $\alpha $ is a countable ordinal. We establish three results:
(1)G is $0$-CLI iff $G=\{1_G\}$;
(2)G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and
(3)G is L-$\alpha $-CLI iff G is locally $\alpha $-CLI, i.e., G contains an open subgroup that is $\alpha $-CLI.
Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that:
(1)$H_\alpha $ is $\alpha $-CLI but not L-$\beta $-CLI for $\beta <\alpha $; and
(2)$G_\alpha $ is $(\alpha +1)$-CLI but not L-$\alpha $-CLI.