Geometry and rigidity of mapping class groups

Abstract

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.