Dimension and rank for mapping class groups

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.Comment: Incorporates referee's suggestions. To appear in Annals of Mathematic

Quasi-isometric classification of non-geometric 3-manifold groups

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.Comment: Minor revision (added footnote in the Introduction

Centroids and the Rapid Decay property in mapping class groups

We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).Comment: v3. Numerous typos fixed and some arguments elucidate

Asymptotic geometry of the mapping class group and Teichmueller space

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Drutu and Sapir; this tree-grading has several consequences including answering a question of Drutu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmueller space, with the Weil-Petersson metric, is delta-hyperbolic. Although for higher complexity surfaces these spaces are not delta-hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmueller space for arbitrary surfaces.Comment: This is the version published by Geometry & Topology on 21 October 200