Geometry & Topology
- Geom. Topol.
- Volume 16, Number 2 (2012), 781-888.
Geometry and rigidity of mapping class groups
We study the large scale geometry of mapping class groups , using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for , namely that groups quasi-isometric to 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 ; a characterization of the image of the curve complex projections map from to ; and a construction of –hulls in , an analogue of convex hulls.
Geom. Topol., Volume 16, Number 2 (2012), 781-888.
Received: 9 April 2010
Revised: 8 February 2012
Accepted: 8 February 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20F36: Braid groups; Artin groups 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F69: Asymptotic properties of groups
Secondary: 57M50: Geometric structures on low-dimensional manifolds 30F60: Teichmüller theory [See also 32G15]
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