We study the outer automorphism group of a right-angled Artin group AΓ with finite defining graph Γ. We construct a subnormal series for Out(AΓ) such that each consecutive quotient is either finite, free-abelian, GL(n, Z) or a Fouxe-Rabinovitch group. The last two types act, respectively, on a symmetric space or a deformation space of trees, so that there is a geometric way of studying each piece. As a consequence we prove that the group Out(AΓ) is type VF (it has a finite index subgroup with a finite classifying space).The main technical work is a study of relative outer automorphism groups of RAAGs and their restriction homomorphisms, refining work of Charney, Crisp and Vogtmann. We show that the images and kernels of restriction homomorphisms are always simpler examples of relative outer automorphism groups of RAAGs. We also give generators for relative automorphism groups of RAAGs, in the style of Laurence's theorem.

We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.

If G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

Abstract. Let G be a real semisimple Lie group with no compact factors and finite centre, and let Λ be an irreducible lattice in G. Suppose that there exists a homomorphism from Λ to the outer automorphism group of a right-angled Artin group A Γ with infinite image. We give a strict upper bound to the real rank of G that is determined by the structure of cliques in Γ. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup T (A Γ ) of Aut(A Γ ). We answer a question of Day relating to the abelianisation of T (A Γ ), and show that T (A Γ ) and its image in Out(A Γ ) are residually torsion-free nilpotent.

Abstract. We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(F n ). This gives restrictions to actions of Aut(F n ) on CAT(0) spaces.