Large-scale rank of Teichmüller space

Abstract

Suppose that $\mathcal{X}$ is either the mapping class group equipped with the word metric or Teichmüller space equipped with either the Teichmüller metric or the Weil–Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that for any large box in ${\mathbb{R}}^n$ there is a standard model of a flat in $\mathcal{X}$ such that the quasi-Lipschitz image of a large sub-box is near the standard flat. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.