{"title":"Boundary rigidity of Gromov hyperbolic spaces","authors":"Hao Liang, Qingshan Zhou","doi":"10.1007/s10711-024-00947-7","DOIUrl":null,"url":null,"abstract":"<p>We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are uniformly perfect. Also, boundary rigidity is shown to be equivalent to being geodesically rich, a concept introduced by Shchur (J Funct Anal 264(3):815–836, 2013).</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometriae Dedicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00947-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are uniformly perfect. Also, boundary rigidity is shown to be equivalent to being geodesically rich, a concept introduced by Shchur (J Funct Anal 264(3):815–836, 2013).
期刊介绍：
Geometriae Dedicata concentrates on geometry and its relationship to topology, group theory and the theory of dynamical systems.
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas. Features of the journal will include:
A fast turn-around time for articles.
Special issues centered on specific topics.
All submitted papers should include some explanation of the context of the main results.
Geometriae Dedicata was founded in 1972 on the initiative of Hans Freudenthal in Utrecht, the Netherlands, who viewed geometry as a method rather than as a field. The present Board of Editors tries to continue in this spirit. The steady growth of the journal since its foundation is witness to the validity of the founder''s vision and to the success of the Editors'' mission.
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