Geometry and rigidity of mapping class groups

Abstract

We study the large scale geometry of mapping class groups $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$, using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$ (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$, namely that groups quasi-isometric to $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$ 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 $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$; a characterization of the image of the curve complex projections map from $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$ to ${\prod }_{Y\subseteq S}\mathcal{C}\left(Y\right)$; and a construction of $\Sigma$–hulls in $\mathcal{ℳ}\mathcal{C}\mathcal{G}\left(S\right)$, an analogue of convex hulls.