Geometry and rigidity of mapping class groups

By Jason A. Behrstock, Bruce Kleiner, Yair N. Minsky, Lee Mosher. 2008

142 citations recorded by the source corpus.

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 left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt 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 the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc17db519a2b2f015ab8fe5f9433768fdba2cbb4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, methodology): “e between points of MCGpSqis coarsely the sum over all subsurfaces of the distances between the projections of these points to the various curve graphs. Meanwhile, the consistency/realization theorem [BKMM12] tells us that tuples with coordinates in the dierent curve graphs that obey \consistency&qu”
Covers and curve complex (2007, methodology): “This has recently been obtained for the mapping class group by Behrstock, Kleiner, Minsky and Mosher [1] and also by Hamenstädt [8].”
Quasiflats in hierarchically hyperbolic spaces (2017, methodology): “llows by applying the same argument to a quasi-inverse of f. 5.2. Mapping class groups. We now use Theorem 5.8 to provide a new proof of quasiisometric rigidity of mapping class groups. Theorem 5.11. [BKMM12] Let X be the the mapping class group of a non-sporadic surface S. Then for any Kthere exist”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, background): “The class of hierarchically hyperbolic spaces consists of metric spaces whose geometry can be recovered, coarsely, from projections onto a specified collection of hyperbolic metric spaces; the axioms governing these spaces and projections are modeled on the relation between the subsurface projection”