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Large-scale rank and rigidity of the Teichmüller metric
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Bowditch, B. H. (2016) Large-scale rank and rigidity of the Teichmüller metric. Journal of Topology, 9 (4). pp. 985-1020. doi:10.1112/jtopol/jtw017 ISSN 1753-8416.
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Official URL: http://dx.doi.org/10.1112/jtopol/jtw017
Abstract
We study the coarse geometry of the Teichmüller space of a compact orientable surface in the Teichmüller metric. We describe when this admits a quasi‐isometric embedding of a euclidean space, or a euclidean half‐space. We prove quasi‐isometric rigidity for Teichmüller space of a surface of complexity at least 2: a result proved independently by Eskin, Masur and Rafi. We deduce that, apart from some well‐known coincidences, the Teichmüller spaces are quasi‐isometrically distinct. We also show that Teichmüller space satisfies a quadratic isoperimetric inequality. A key ingredient for proving these results is the fact that Teichmüller space admits a ternary operation, natural up to bounded distance, which endows the space with the structure of a coarse median space whose rank is equal to the complexity of the surface. From this, one can also deduce that any asymptotic cone is bilipschitz equivalent to a CAT(0) space, and so, in particular, is contractible.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Teichmüller spaces, Class groups (Mathematics), Isoperimetric inequalities, Hyperboloid structures | ||||||
Journal or Publication Title: | Journal of Topology | ||||||
Publisher: | Oxford University Press | ||||||
ISSN: | 1753-8416 | ||||||
Official Date: | 12 August 2016 | ||||||
Dates: |
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Volume: | 9 | ||||||
Number: | 4 | ||||||
Page Range: | pp. 985-1020 | ||||||
DOI: | 10.1112/jtopol/jtw017 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 18 May 2016 | ||||||
Date of first compliant Open Access: | 18 May 2016 | ||||||
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