In 2007, Charney–Crisp–Vogtmann introduced restriction and projection homomorphisms for automorphisms of RAAGs. These homomorphisms make it possible to prove many results about Out(AΓ) (the outer automorphism group of a RAAG) by inductive methods. However, this method has a key limitation: the images of restriction homomorphisms are difficult to describe.
In joint work with Ric Wade, we have overcome this difficulty. We introduce a generalization of Out(AΓ): the relative outer automorphism group of AΓ. This is a kind of conjugacy stabilizer of certain subgroups of AΓ. Aside from a few well-understood examples, every relative Out(AΓ) has a nontrivial restriction homomorphism, and the image and kernel of this homomorphism are simpler examples of relative outer automorphism groups. This makes the inductive method more powerful. As sample results, we show that Out(AΓ) has type VF (this is a strengthening of finite presentability) and we give a general description of Out(AΓ) as an iterated extension of more familiar groups.