Given a group G, we say all finite groups are involved in G if for each finite group H there is a finite index subgroup of G which admits a surjection onto H. From the subgroup congruence property, it is known that the groups GL(n,Z) do not have every finite group involved for n>2. Meanwhile, the representations of Out(Fn) given by Grunewald and Lubotzky imply that these groups do have all finite groups involved. We will describe conditions on the defining graph of a RAAG that are necessary and sufficient to determine when it's outer automorphism group has this property. The same criterion also holds for other properties, such as SQ-universality, or having infinite dimensional second bounded cohomology.
This is joint work with V. Guirardel.