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

Yi I Huang (1) , Athanase Papadopoulos (2, 3)

1 Yau Mathematical Sciences Center, Tsinghua University,
2 IRMA - Institut de Recherche Mathématique Avancée
3 GREAM - Groupe de recherches expérimentales sur l'acte musical

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 not larger than 4 Arcsinh(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. The final version of this paper will appear in Geometriae Dedicata.

Mots clés: 53C70; 53C23; stretch map; stretch path; arc metric; Teichmüller space; hyperbolic surface; Lipschitz metric; curve metric; Thurston metric; partial stretch path; geodesic; Gromov hyperbolicity; Gromov hyperbolicity AMS classification-32G15; 30F60; 30F10; Gromov hyperbolicity AMS classification.-32G15; 53C70

Domaines: Topologie géométrique [math.GT]

Yi I Huang, Athanase Papadopoulos. Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space. 2019. ⟨hal-02282383v2⟩