We study the coarse geometry of the Teichmüller space of a compact surface in the Teichmüller metric. We show that this admits a ternary operation, natural up to bounded distance, which endows the space with the structure of a coarse median space whose rank is equal to the complexity of the surface. We deduce that Teichmüller space satisfies a coarse quadratic isoperimetric inequality. We describe when it admits a quasi-isometric embedding of a euclidean space, or a euclidean half-space. We give a weak form of quasi-isometric rigidity for Teichmüller space, and deduce that, apart from some well known coincidences, the Teichmüller spaces are quasi-isometrically distinct. We show that any asymptotic cone is bilipschitz equivalent to a CAT(0) space, and so in particular, is contractible.
Preprint, Warwick/Tokyo, October 2014.
This paper has been superseded by the paper:
``Large-scale rank and rigidiy of the Teichmüller metric''.