Geometry & Topology

Geometry and rigidity of mapping class groups

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 $∏Y⊆SC(Y)$; and a construction of $Σ$–hulls in $ℳCG(S)$, an analogue of convex hulls.

Article information

Source
Geom. Topol., Volume 16, Number 2 (2012), 781-888.

Dates
Revised: 8 February 2012
Accepted: 8 February 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732409

Digital Object Identifier
doi:10.2140/gt.2012.16.781

Mathematical Reviews number (MathSciNet)
MR2928983

Zentralblatt MATH identifier
1281.20045

Citation

Behrstock, Jason; Kleiner, Bruce; Minsky, Yair; Mosher, Lee. Geometry and rigidity of mapping class groups. Geom. Topol. 16 (2012), no. 2, 781--888. doi:10.2140/gt.2012.16.781. https://projecteuclid.org/euclid.gt/1513732409

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