  We study the large scale geometry of the mapping class group, MCG. Our main
result is that for any asymptotic cone of MCG, the maximal dimension of locally
compact subsets coincides with the maximal rank of free abelian subgroups of
MCG. An application is an affirmative solution to Brock-Farb's Rank Conjecture
which asserts that MCG has quasi-flats of dimension N if and only if it has a
rank N free abelian subgroup. We also compute the maximum dimension of
quasi-flats in Teichmuller space with the Weil-Petersson metric.
