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

Abstract

We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ 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 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 $\mathrm{SOut}(F_n)$ can act non-trivially.

16 pages v.2 Minor changes. Final version

References in corpus (1)

• Hochschild-Pirashvili homology on suspensions and representations of $Out(F_n)$

Cited by in corpus (3)

• Relative automorphism groups of right-angled Artin groups
• On the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$
• The action of matrix groups on aspherical manifolds