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

Abstract

We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmuller 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 Teichmuller 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 $$\le 4{\text {arcsinh}}(1)$$ , but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmuller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.