Geometry and rigidity of mapping class groups

We study the large scale geometry of mapping class groups MCG.(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG.(S) (outside a few sporadic cases) is a bounded distance away from a leftmultiplication, and as a consequence obtain quasi-isometric rigidity for MCG.(S), namely that groups quasi-isometric to MCG.(S) are equivalent to it up to extraction of finite-index subgroups and quotients with finite kernel. (The latter theorem was proved by Hamenstädt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG.(S) a characterization of theQ image of the curve complex projections map from MCG.(S) to π _Y⊂S C.Y and a construction of ∑-hulls in MCG.(S) an analogue of convex hulls.