Suppose that 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 there is a standard model of a flat in 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.
Alex Eskin. Howard Masur. Kasra Rafi. "Large-scale rank of Teichmüller space." Duke Math. J. 166 (8) 1517 - 1572, 1 June 2017. https://doi.org/10.1215/00127094-0000006X