  We study Thurston's Lipschitz and curve metrics, as well as the arc metric on
the Teichmueller 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 Teichmueller 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
=< 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
Teichmueller spaces of closed hyperbolic surfaces, and use these geodesics to
show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov
hyperbolicity.
