# 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.

## Authors

Jason A. Behrstock

Department of Mathematics
University of Utah
Salt Lake City, UT 84112
United States

Yair N. Minsky

Department of Mathematics
Yale University
New Haven, CT 06520
United States