Geometry and rigidity of mapping class groups

Behrstock, Jason¹ ; Kleiner, Bruce² ; Minsky, Yair³ ; Mosher, Lee⁴

¹ The Graduate Center and Lehman College, CUNY, New York NY 10016, USA
² Department of Mathematics, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York NY 10012-1185, USA
³ Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven CT 06520-8283, USA
⁴ Department of Mathematics and Computer Science, Rutgers University Newark, Newark NJ 07102, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 781-888.

Résumé

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.

Détail

DOI : 10.2140/gt.2012.16.781

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

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