On the number of outer automorphisms of the automorphism group of a right-angled Artin group

TL;DR

This paper demonstrates that the size of the outer automorphism group of the automorphism group of right-angled Artin groups is unbounded, contrasting with known bounds for free and free abelian groups, using explicit examples and austere graphs.

Contribution

It proves there is no uniform bound on |Out(Aut(A))| for all right-angled Artin groups and introduces austere graphs as a new tool.

Findings

No uniform upper bound on |Out(Aut(A))| for all right-angled Artin groups.

Explicit examples show unbounded automorphism groups.

Introduces austere graphs to aid in analysis.

Abstract

We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges over all right-angled Artin groups. This is in contrast with the cases where A is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))| < 5. We also prove the analogous theorem for Out(Out(A)). We establish our results by giving explicit examples; one useful tool is a new class of graphs called austere graphs.