{"title":"Large-scale rank and rigidity of the Weil–Petersson metric","authors":"B. Bowditch","doi":"10.4171/GGD/557","DOIUrl":null,"url":null,"abstract":"We study the large-scale geometry of Weil-Petersson space, that is, Teichmüller space equipped with the Weil-Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space, we recover a result of Eskin, Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the WeilPetersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock, Weil-Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"14 1","pages":"607-652"},"PeriodicalIF":0.6000,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Groups Geometry and Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/GGD/557","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 11
Abstract
We study the large-scale geometry of Weil-Petersson space, that is, Teichmüller space equipped with the Weil-Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space, we recover a result of Eskin, Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the WeilPetersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock, Weil-Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that
期刊介绍:
Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.
Topics covered include:
geometric group theory;
asymptotic group theory;
combinatorial group theory;
probabilities on groups;
computational aspects and complexity;
harmonic and functional analysis on groups, free probability;
ergodic theory of group actions;
cohomology of groups and exotic cohomologies;
groups and low-dimensional topology;
group actions on trees, buildings, rooted trees.