We determine the precise conditions under which SOut(Fnβ), the
unique index two subgroup of Out(Fnβ), can act non-trivially via
outer automorphisms on a RAAG whose defining graph has fewer than 21β(2nβ) 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 SOut(Fnβ) can act
non-trivially.Comment: 16 pages v.2 Minor changes. Final versio