Dimension and rank for mapping class groups. (English) Zbl 1280.57015

Summary: 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. (Hamenstädt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmüller space with the Weil-Petersson metric.


57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
20F69 Asymptotic properties of groups
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