Abstract
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.
Citation
Jason Behrstock. Bruce Kleiner. Yair Minsky. Lee Mosher. "Geometry and rigidity of mapping class groups." Geom. Topol. 16 (2) 781 - 888, 2012. https://doi.org/10.2140/gt.2012.16.781
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