LARGE-SCALE RANK OF TEICHMULLER SPACE

Suppose that X is either the mapping class group equipped with the word metric or Teichmuller space equipped with either the Teichmuller 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 R-n there is a standard model of a flat in 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.