Dimension and rank for mapping class groups

Abstract

We study the large scale geometry of the mapping class group, $\mathcal{MCG}(S)$. Our main result is that for any asymptotic cone of $\mathcal{MCG}(S)$, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of $\mathcal{MCG}(S)$. An application is a proof of Brock-Farb’s Rank Conjecture which asserts that $\mathcal{MCG}(S)$ has quasi-flats of dimension $N$ if and only if it has a rank $N$ free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric.