Harmonic diffeomorphisms and Teichmüller theory.
Homomorphisms of constant stretch between Möbius groups.
Hyperbolic spaces in Teichmüller spaces
We prove, for any , that there is a closed connected orientable surface so that the hyperbolic space almost-isometrically embeds into the Teichmüller space of , with quasi-convex image lying in the thick part. As a consequence, quasi-isometrically embeds in the curve complex of .