We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb-Kaimanovich-Masur rigidity theorem that states that MCG(S) does not contain a higher rank lattice as a subgroup.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper4.abs.htm

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasitrees. The groups we can handle include non-elementary (relatively) hyperbolic groups, CAT (0) groups with rank 1 elements, mapping class groups and Out(F n ). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.

Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal coverM is a higher rank symmetric space iff H 2 b (M ; R) → H 2 (M ; R) is injective (and otherwise the kernel is infinite-dimensional). This is the converse of a theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT (0) spaces and contain rank 1 elements.

Abstract. A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left and right invariant word metrics.Examples of bicombable functions on word-hyperbolic groups include (1) homomorphisms to Z (2) word length with respect to a finite generating set (3) most known explicit constructions of quasimorphisms (e.g. the EpsteinFujiwara counting quasimorphisms) We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: If φ n is the value of φ on a random element of word length n (in a certain sense), there are E and σ for which there is convergence in the sense of distribution n −1/2 (φ n − nE) → N (0, σ), where N (0, σ) denotes the normal distribution with standard deviation σ. As a corollary, we show that if S 1 and S 2 are any two finite generating sets for G, there is an algebraic number λ 1,2 depending on S 1 and S 2 such that almost every word of length n in the S 1 metric has word length n · λ 1,2 in the S 2 metric, with error of size O( √ n).

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