Vastness properties of automorphism groups of RAAGs

Outer automorphism groups of RAAGs, denoted $Out(A_\Gamma)$, interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $GL_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality, virtually having an infinite dimensional space of homogeneous quasimorphisms, and not being boundedly generated. We give a neccessary and sufficient condition in terms of the defining graph $\Gamma$ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning $Out(A_\Gamma)$ will either satisfy all four, or none. In proving this result, we describe conditions on $\Gamma$ that imply $Out(A_\Gamma)$ is large. Techniques used in this work are then applied to the case of McCool groups, defined as subgroups of $Out(F_n)$ that preserve a given family of conjugacy classes. In particular we show that any McCool group that is not virtually abelian virtually maps onto all finite groups, is SQ-universal, is not boundedly generated, and has a finite index subgroup whose space of homogeneous quasimorphisms is infinite dimensional.