Geometry & Topology 16 (2012) 781–888

DOI: 10.2140/gt.2012.16.781

Abstract

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

Primary: 20F34, 20F36, 20F65, 20F69

Secondary: 30F60, 57M50

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