Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Pub Date : 2024-09-18 DOI:10.1017/etds.2024.55
DONGRYUL M. KIM, HEE OH
{"title":"Non-concentration property of Patterson–Sullivan measures for Anosov subgroups","authors":"DONGRYUL M. KIM, HEE OH","doi":"10.1017/etds.2024.55","DOIUrl":null,"url":null,"abstract":"Let <jats:italic>G</jats:italic> be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline1.png\"/> <jats:tex-math> $\\Gamma &lt;G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with respect to a parabolic subgroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline2.png\"/> <jats:tex-math> $P_\\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove that any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline3.png\"/> <jats:tex-math> $\\Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Patterson–Sullivan measure charges no mass on any proper subvariety of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline4.png\"/> <jats:tex-math> $G/P_\\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. More generally, we prove that for a Zariski dense <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline5.png\"/> <jats:tex-math> $\\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-transverse subgroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline6.png\"/> <jats:tex-math> $\\Gamma &lt;G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline7.png\"/> <jats:tex-math> $(\\Gamma , \\psi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Patterson–Sullivan measure charges no mass on any proper subvariety of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline8.png\"/> <jats:tex-math> $G/P_\\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, provided the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline9.png\"/> <jats:tex-math> $\\psi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Poincaré series of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000555_inline10.png\"/> <jats:tex-math> $\\Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","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/etds.2024.55","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract

Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $ , we prove that any $\Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ . More generally, we prove that for a Zariski dense $\theta $ -transverse subgroup $\Gamma <G$ , any $(\Gamma , \psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ , provided the $\psi $ -Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.
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阿诺索夫子群的帕特森-沙利文度量的非集中特性
让 G 是一个连通的半简单实代数群。对于相对于抛物子群 $P_\theta $ 的扎里斯基密集阿诺索夫子群 $\Gamma <G$,我们证明任何 $\Gamma $ -帕特森-沙利文度量在 $G/P_\theta $ 的任何适当子变上都不带质量。更广义地说,我们证明了对于一个扎里斯基密集的 $\theta $ -反子群 $\Gamma <G$ ,任何 $(\Gamma , \psi )$ -帕特森-沙利文度量在 $G/P_\theta $ 的任何适当子变上都不收取质量,只要 $\Gamma $ 的 $\psi $ -波因卡雷数列在其收敛的上位发散。特别是,我们的结果也适用于相对阿诺索夫子群。
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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