A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC

LONGYUN DING, XU WANG
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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 &lt;\\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 &lt;\\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 Abstract Image$\alpha $-CLI and L-Abstract Image$\alpha $-CLI where Abstract Image$\alpha $ is a countable ordinal. We establish three results:

  1. (1) G is Abstract Image$0$-CLI iff Abstract Image$G=\{1_G\}$;

  2. (2) G is Abstract Image$1$-CLI iff G admits a compatible complete two-sided invariant metric; and

  3. (3) G is L-Abstract Image$\alpha $-CLI iff G is locally Abstract Image$\alpha $-CLI, i.e., G contains an open subgroup that is Abstract Image$\alpha $-CLI.

Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups Abstract Image$G_\alpha $ and Abstract Image$H_\alpha $ for Abstract Image$\alpha <\omega _1$, such that:

  1. (1) Abstract Image$H_\alpha $ is Abstract Image$\alpha $-CLI but not L-Abstract Image$\beta $-CLI for Abstract Image$\beta <\alpha $; and

  2. (2) Abstract Image$G_\alpha $ is Abstract Image$(\alpha +1)$-CLI but not L-Abstract Image$\alpha $-CLI.

允許相容完整左不變度量的非archimedean拋光群上的階級結構
在这篇文章中,我们介绍了一类非archimedean波兰群的层次结构,它们承认一个兼容的完全左不变度量。我们用 $\alpha $-CLI 和 L-$\alpha $-CLI 表示这个层次,其中 $\alpha $ 是一个可数序号。我们建立了三个结果:(1)如果 $G=\{1_G\}$ 是 $0$-CLI,则 G 是 $0$-CLI;(2)如果 G 允许一个兼容的完整双面不变度量,则 G 是 $1$-CLI;(3)如果 G 是局部 $\alpha $-CLI,即 G 包含一个开放子群,而这个开放子群在 G 的局部是 $\alpha $-CLI,则 G 是 L-$\alpha $-CLI、随后,我们通过为$\alpha <\omega _1$构造非拱顶的CLI波兰群$G_\alpha $和$H_\alpha $来证明这个层次结构是合适的,这样的话:(1) $H_\alpha $ 是 $\alpha $-CLI 但不是 L-$\beta $-CLI for $\beta <\alpha $;(2) $G_\alpha $ 是 $(\alpha +1)$-CLI 但不是 L-$\alpha $-CLI。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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