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

Abstract

We determine the precise conditions under which $SOut\left(F_n\right)$, the unique index-two subgroup of $Out\left(F_n\right)$, can act nontrivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac{1}{2}\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ 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 nontrivial 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 $SOut\left(F_n\right)$ can act nontrivially.