Geometry and rigidity of mapping class groups

Jason Behrstock, Bruce Kleiner, Yair Minsky, Lee Mosher

Research output: Contribution to journalArticle

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 leftmultiplication, 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 theQ image of the curve complex projections map from MCG.(S) to π Y⊂S C.Y and a construction of ∑-hulls in MCG.(S) an analogue of convex hulls.

Original languageEnglish (US)
Pages (from-to)781-888
Number of pages108
JournalGeometry and Topology
Volume16
Issue number2
DOIs
StatePublished - 2012

ASJC Scopus subject areas

  • Geometry and Topology

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