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COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

Published online by Cambridge University Press:  26 May 2020

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

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.

Type
Differential Geometry and Geometric Analysis
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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