Geometry and rigidity of mapping class groups

TL;DR

This paper investigates the large-scale geometry of mapping class groups, establishing quasi-isometric rigidity and structural properties using hyperbolicity of curve complexes and asymptotic cone analysis.

Contribution

It proves that most self quasi-isometries of MCG(S) are close to group translations and characterizes the geometric structure of MCG(S) through new structural results.

Findings

Any self quasi-isometry of MCG(S) is close to a left-multiplication.

Groups quasi-isometric to MCG(S) are virtually isomorphic.

Structural descriptions of the asymptotic cone and curve complex projections.

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 virtually equal to it. (The latter theorem was proved by Hamenstadt 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 projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.