Let F n be the free group on n generators. Define IA n to be group of automorphisms of F n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of IA n .

No abstract

Let G be a compact Lie group. Consider the variety Hom(Z k , G) of representations of Z k into G. We can see this as a based space by taking as base point the trivial representation 1. The goal of this paper is to prove that π 1 (Hom(Z k , G)) is naturally isomorphic to π 1 (G) k .

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.

Abstract. We show that for k ≥ 3, given any matrix in GL(k, Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z k is the given matrix.

We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup H of Out(FN ) quasi-isometrically embeds into the complex of free factors via the orbit map, then the limit set of H in the boundary of Outer space consists of trees that are uniquely ergodic and have uniquely ergodic dual lamination. Second, we describe the Poisson boundary for random walks coming from distributions with finite first moment with respect to the word metric on Out(FN ): almost every sample path converges to a tree that is uniquely ergodic and that has a uniquely ergodic dual lamination, and the corresponding hitting measure on the boundary of Outer space is the Poisson boundary. This improves a recent result of Horbez. We also obtain sublinear tracking of sample paths with Lipschitz geodesic rays.

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus g which fill and pairwise intersect at most K ≥ 1 times is 2 √ g/ √ K as g → ∞ . We then bound from below the cardinality of a filling set of systoles by g/ log(g). This illustrates that the topological condition that a set of curves pairwise intersect at most once is quite far from the geometric condition that such a set of curves can arise as systoles.arXiv: 0909.1966v2 [math.GT]

We prove that every non-positively curved, locally symmetric manifold M of finite volume contains a compact set K such that no periodic maximal flat in M can be homotoped out of K.