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COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

Part of: Low-dimensional topology Riemann surfaces

Published online by Cambridge University Press:  26 May 2020

DAVID DUMAS Open the ORCID record for DAVID DUMAS [Opens in a new window] ,
ANNA LENZHEN ,
KASRA RAFI Open the ORCID record for KASRA RAFI [Opens in a new window]  and
JING TAO
DAVID DUMAS
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, USA; david@dumas.io
ANNA LENZHEN
Affiliation:
Department of Mathematics, University of Rennes, Rennes, France; anna.lenzhen@univ-rennes1.fr
KASRA RAFI
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada; rafi@math.toronto.edu
JING TAO
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK, USA; jing@ou.edu

Abstract

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We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F60: Teichmüller theory
Type
Differential Geometry and Geometric Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e28
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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