{"title":"CAT(0)可接纳群的次线性莫尔斯边界","authors":"Hoang Thanh Nguyen, Yulan Qing","doi":"10.1515/jgth-2023-0145","DOIUrl":null,"url":null,"abstract":"We show that if 𝐺 is an admissible group acting geometrically on a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>CAT</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0001.png\" /> <jats:tex-math>\\operatorname{CAT}(0)</jats:tex-math> </jats:alternatives> </jats:inline-formula> space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mo lspace=\"0em\" rspace=\"0em\">∂</m:mo> <m:mi>κ</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>ν</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0002.png\" /> <jats:tex-math>(\\partial_{\\kappa}G,\\nu)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a model for the Poisson boundary of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0003.png\" /> <jats:tex-math>(G,\\mu)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sublinearly Morse boundary of CAT(0) admissible groups\",\"authors\":\"Hoang Thanh Nguyen, Yulan Qing\",\"doi\":\"10.1515/jgth-2023-0145\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that if 𝐺 is an admissible group acting geometrically on a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>CAT</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0145_ineq_0001.png\\\" /> <jats:tex-math>\\\\operatorname{CAT}(0)</jats:tex-math> </jats:alternatives> </jats:inline-formula> space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:msub> <m:mo lspace=\\\"0em\\\" rspace=\\\"0em\\\">∂</m:mo> <m:mi>κ</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>ν</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0145_ineq_0002.png\\\" /> <jats:tex-math>(\\\\partial_{\\\\kappa}G,\\\\nu)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a model for the Poisson boundary of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0145_ineq_0003.png\\\" /> <jats:tex-math>(G,\\\\mu)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2023-0145\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0145","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Sublinearly Morse boundary of CAT(0) admissible groups
We show that if 𝐺 is an admissible group acting geometrically on a CAT(0)\operatorname{CAT}(0) space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary (∂κG,ν)(\partial_{\kappa}G,\nu) is a model for the Poisson boundary of (G,μ)(G,\mu), where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.
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