Geometry and rigidity of mapping class groups

Details Geometry and rigidity of mapping class groups Geometry and rigidity of mapping class groups

2008-01-14

arxiv.org/abs/0801.2006v4

Mathematics

Geometric Topology

Version 4, 98 pages. Cleaned up some notation involving the partial order on subsurfaces (at the end of section 4).

Scientific paper

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.

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