  Let X be quasi-isometric to either the mapping class group equipped with the
word metric, or to 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 the quasi-Lipschitz image in
X of a box in R^n is locally near a standard model of a flat in X. 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.
