Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

Yi Huang YMSC, Tsinghua University Athanase Papadopoulos IRMA, University of Strasbourg

Geometric Analysis and Geometric Topology arXiv subject: Geometric Topology (math.GT) mathscidoc:2206.15002

Geometriae Dedicata, 214, 465-488, 2021.4
We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is ≤4arcsinh(1), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.
Lipschitz homeomorphism, Thurston metric, hyperbolic surfaces, Teichmueller theory
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@inproceedings{yi2021optimal,
  title={Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space},
  author={Yi Huang, and Athanase Papadopoulos},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220608152247481327344},
  booktitle={Geometriae Dedicata},
  volume={214},
  pages={465-488},
  year={2021},
}
Yi Huang, and Athanase Papadopoulos. Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space. 2021. Vol. 214. In Geometriae Dedicata. pp.465-488. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220608152247481327344.
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