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Geometry and rigidity of mapping class groups

Jason Behrstock, Bruce Kleiner, Yair Minsky and Lee Mosher

Geometry & Topology 16 (2012) 781–888

We study the large scale geometry of mapping class groups CG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of CG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for CG(S), namely that groups quasi-isometric to CG(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 CG(S); a characterization of the image of the curve complex projections map from CG(S) to Y SC(Y ); and a construction of Σ–hulls in CG(S), an analogue of convex hulls.

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
Received: 9 April 2010
Revised: 8 February 2012
Accepted: 8 February 2012
Published: 15 May 2012
Proposed: Benson Farb
Seconded: David Gabai, Danny Calegari
Jason Behrstock
The Graduate Center and Lehman College, CUNY
New York NY 10016
Bruce Kleiner
Department of Mathematics
Courant Institute of Mathematical Sciences
251 Mercer Street
New York NY 10012-1185
Yair Minsky
Department of Mathematics
Yale University
10 Hillhouse Ave
New Haven CT 06520-8283
Lee Mosher
Department of Mathematics and Computer Science
Rutgers University Newark
Newark NJ 07102