Geometry and rigidity of mapping class groups

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 the image of the curve complex projections map from MCG.S/ to Q Y ÂS C.Y /; and a construction of †-hulls in MCG.S /, an analogue of convex hulls.

The geometry of right-angled Artin subgroups of mapping class groups

“…We need only ensure that this progress accumulates (that is, we need to avoid cancelation of local distances). This is verified by Theorem 5.2, which relates a partial order on the set of syllables in a minimal length representative for an element of the right-angled Artin group (see Section 4) with the partial order from [3] on the set of subsurfaces "between" a marking and its image under the associated mapping class (see Section 3.4). The details of the proof of Theorem 5.2 are carried out in Section 5, followed by the proof of Theorem 2.1.…”

“…In this case, we say that X and Y are overlapping, and write X Y if X ⊆ Y and Y ⊆ X. One can check that this notion of overlapping agrees with that defined in [3], which is to say that X Y if and only if some component of ∂X cannot be isotoped disjoint from Y and some component of ∂Y cannot be isotoped disjoint from X.…”

“…Our proof of Theorem 2.1 uses results from [33], [6], [38] and [3]. The main construction we will use is that of subsurface projection, which we now briefly recall.…”

“…In [3], Behrstock, Kleiner, Minsky and Mosher defined a partial order on Ω(K, µ, µ ) (for K sufficiently large) which is closely related to the time-order constructed in [33] (see also [4]). However, as is noted in [3], while the time-order in [33] (which is defined on geodesics in hierarchies) requires a fair amount of the hierarchy machinery to describe it, the partial order on Ω(K, µ, µ ) is completely elementary.…”

“…However, as is noted in [3], while the time-order in [33] (which is defined on geodesics in hierarchies) requires a fair amount of the hierarchy machinery to describe it, the partial order on Ω(K, µ, µ ) is completely elementary. As this is the basic tool we will use, we include the construction and verification of the necessary properties of this partial order, for the sake of completeness.…”