Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

TL;DR

This paper constructs the first known examples of infinite subgroups within the outer automorphism group of certain right-angled Artin groups, revealing new complex subgroup structures and disproving previous conjectures.

Contribution

It introduces focused graphs to exhibit infinite projective linear groups as subgroups of Out(Out(A_Gamma)), expanding understanding of automorphism groups of right-angled Artin groups.

Findings

Existence of infinite subgroups in Out(Out(A_Gamma)) for certain RAAGs

Introduction of focused graphs enabling new subgroup constructions

Disproof of a previous conjecture regarding finite order members

Abstract

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A_Gamma)).