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We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup G of MCG de ning an extension 1 ! 1(S) ! ΓG ! G ! 1, we prove that if ΓG is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free(More)
Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmüller space Teich(S) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the(More)
We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much(More)
We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly’s 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encoded in a property that we introduce called “property FAr”, which(More)
Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod±(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the(More)
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n-pointed curves; •(More)