{"title":"ONE-DIMENSIONAL SUBGROUPS AND CONNECTED COMPONENTS IN NON-ABELIAN p-ADIC DEFINABLE GROUPS","authors":"WILLIAM JOHNSON, NINGYUAN YAO","doi":"10.1017/jsl.2024.31","DOIUrl":null,"url":null,"abstract":"<p>We generalize two of our previous results on abelian definable groups in <span>p</span>-adically closed fields [12, 13] to the non-abelian case. First, we show that if <span>G</span> is a definable group that is not definably compact, then <span>G</span> has a one-dimensional definable subgroup which is not definably compact. This is a <span>p</span>-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if <span>G</span> is a group definable over the standard model <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513074612283-0489:S0022481224000318:S0022481224000318_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {Q}_p$</span></span></img></span></span>, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513074612283-0489:S0022481224000318:S0022481224000318_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G^0 = G^{00}$</span></span></img></span></span>. As an application, definably amenable groups over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513074612283-0489:S0022481224000318:S0022481224000318_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {Q}_p$</span></span></img></span></span> are open subgroups of algebraic groups, up to finite factors. We also prove that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513074612283-0489:S0022481224000318:S0022481224000318_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G^0 = G^{00}$</span></span></img></span></span> when <span>G</span> is a definable subgroup of a linear algebraic group, over any model.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-29","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.2024.31","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
We generalize two of our previous results on abelian definable groups in p-adically closed fields [12, 13] to the non-abelian case. First, we show that if G is a definable group that is not definably compact, then G has a one-dimensional definable subgroup which is not definably compact. This is a p-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if G is a group definable over the standard model $\mathbb {Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb {Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when G is a definable subgroup of a linear algebraic group, over any model.