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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 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.

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
References
Publication
Received: 9 April 2010
Revised: 8 February 2012
Accepted: 8 February 2012
Published: 15 May 2012
Proposed: Benson Farb
Seconded: David Gabai, Danny Calegari
Authors
Jason Behrstock
The Graduate Center and Lehman College, CUNY
New York NY 10016
USA
http://comet.lehman.cuny.edu/behrstock
Bruce Kleiner
Department of Mathematics
Courant Institute of Mathematical Sciences
251 Mercer Street
New York NY 10012-1185
USA
http://math.nyu.edu/~bkleiner/
Yair Minsky
Department of Mathematics
Yale University
10 Hillhouse Ave
New Haven CT 06520-8283
USA
http://www.math.yale.edu/users/yair
Lee Mosher
Department of Mathematics and Computer Science
Rutgers University Newark
Newark NJ 07102
USA