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

TL;DR

This paper characterizes when the subgroup of outer automorphisms of free groups can act non-trivially on small right-angled Artin groups, revealing limitations based on graph size and representation theory.

Contribution

It provides the first precise conditions for non-trivial actions of $ ext{SOut}(F_n)$ on RAAGs with small defining graphs and establishes bounds on such actions.

Findings

Non-trivial actions require the defining graph to have at least half the binomial coefficient of n choose 2 vertices.

The outer automorphism group of a RAAG cannot act faithfully on a RAAG with fewer vertices than its own.

Minimal dimensions of non-trivial linear representations of certain congruence quotients are determined.

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 \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 $\mathrm{SOut}(F_n)$ can act non-trivially.