Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

Abstract

A right-angled Artin group is a group that admits a finite presentation where the only relations are commutators between two generators. We prove by giving an explicit construction that every right-angled Artin group occurs as a finite-index subgroup of the outer automorphism group of another right-angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to (Z/2Z)^N for some N. For these constructions, we use the group of pure symmetric outer automorphisms, a subgroup of the outer automorphism group of a right-angled Artin group. Moreover, we need two conditions by Day–Wade and Wade–Brück about when this group is a right-angled Artin group and when it has finite index.