Dimension and rank for mapping class groups

• 成果类型: Article
• 署名作者: Behrstock, Jason A.; Minsky, Yair N.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.167.1055
• 发表日期: 2008
• 页码: 1055-1077
• 关键词: teichmuller space; asymptotic cones; geometry; HYPERBOLICITY; complex

摘要

We study the large scale geometry of the mapping class group, MCG(S). Our main result is that for any asymptotic cone of MCG(S), the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S). An application is a proof of Brock-Farb's Rank Conjecture which asserts that 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.