Outer actions of $\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$ on small right-angled Artin groups

We determine the precise conditions under which $\mathrm{S}\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$, the unique index two subgroup of $\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2 $ vertices. We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph. Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $\mathrm{S}\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$ can act non-trivially.