#### Volume 16, issue 2 (2012)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
Geometry and rigidity of mapping class groups

### Jason Behrstock, Bruce Kleiner, Yair Minsky and Lee Mosher

Geometry & Topology 16 (2012) 781–888
##### Abstract

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

##### Keywords
mapping class group, quasi-isometric rigidity, qi rigidity, curve complex, complex of curves, MCG, asymptotic cone
##### Mathematical Subject Classification 2010
Primary: 20F34, 20F36, 20F65, 20F69
Secondary: 57M50, 30F60