Dimension and rank for mapping class groups

Details Dimension and rank for mapping class groups Dimension and rank for mapping class groups

2005-12-15

arxiv.org/abs/math/0512352v4

Annals of Math, 167, (2008), 1055-1077

Mathematics

Geometric Topology

Incorporates referee's suggestions. To appear in Annals of Mathematics

Scientific paper

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.

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