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In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces and of CAT(0)--groups. We show that, for mapping class groups of surfaces, these functions exhibit phase transitions at the rank (as measured by thrice the genus plus the number of punctures minus 3). We also provide inductive constructions of CAT(0)--spaces with co-compact group actions, for which the divergence below the rank is (exactly) a polynomial function of our choice, with degree arbitrarily large compared to the dimension.
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We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.
In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called round and unfolded, provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer--Fleming inequality for finitely generated groups, the construction of examples of $CAT(0)$--groups with higher dimensional divergence equivalent to $x^d$ for every degree d [arXiv:1305.2994], and a proof of the fact that for bi-combable groups the filling function above the quasi-flat rank is asymptotically linear [Behrstock-Drutu].
We survey recent developments on mapping class groups of surfaces of infinite topological type.
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.
We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
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