TY - JOUR

T1 - Geometry and rigidity of mapping class groups

AU - Behrstock, Jason

AU - Kleiner, Bruce

AU - Minsky, Yair

AU - Mosher, Lee

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84863485944&partnerID=8YFLogxK

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U2 - 10.2140/gt.2012.16.781

DO - 10.2140/gt.2012.16.781

M3 - Article

AN - SCOPUS:84863485944

SN - 1364-0380

VL - 16

SP - 781

EP - 888

JO - Geometry and Topology

JF - Geometry and Topology

IS - 2

ER -