Geometry & Topology

Geometry and rigidity of mapping class groups

Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher

Full-text: Open access

Abstract

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

Article information

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

Dates
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
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

Subjects
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]

Keywords
mapping class group quasi-isometric rigidity qi rigidity curve complex complex of curves MCG asymptotic cone

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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