LAUSR

We study the outer automorphism group $Out(A_\Gamma)$ of the right-angled Artin group (RAAG) $A_\Gamma$ with defining graph $\Gamma$. A relative automorphism group is the stabilizer of a set of subgroups up to conjugacy; we consider those subgroups of $Out(A_\Gamma)$ preserving a set of special subgroups. We prove that for every $\Gamma$, the group $Out(A_\Gamma)$ is of type VF (it has a finite index subgroup with a finite classifying space). We prove the existence of these classifying spaces inductively, using similar spaces for relative outer automorphism groups as intermediate steps. To do this, we study restriction homomorphisms on relative outer automorphism groups of RAAGs, refining a technique of Charney 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.