Hyperbolic spaces in Teichmüller spaces

Abstract

We prove, for any $n$, that there is a closed connected orientable surface $S$ so that the hyperbolic space ${\mathbb{H}}^n$ almost-isometrically embeds into the Teichmüller space of $S$, with quasi-convex image lying in the thick part. As a consequence, ${\mathbb{H}}^n$ quasi-isometrically embeds in the curve complex of $S$.