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We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichm\"uller space of the torus equipped with hyperbolic metrics with one boundary component of fixed length. We construct natural Lipschitz maps between two such hyperbolic surfaces that generalize Thurston's stretch maps. The construction is based on maps between ideal Saccheri quadrilaterals. We prove the following: (1) On the Teichmuller space of the torus with one boundary component, the Lipschitz metric and the curve metric coincide and give a geodesic metric. (2) On the same Teichmuller space, the arc metric and the curve metrics coincide when the length of the boundary component is $\leq 4\operatorname{arcsinh}(1)$, but differ when the boundary length is large. We obtain several applications of this construction, including results on the Teichmuller spaces of closed hyperbolic surfaces: we construct novel Thurston geodesics and use them in particular to show that the sum-symmetrization of the Thurston metric is not Gromov hyperbolic.

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