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

Abstract

We prove 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 for some . For these, we give explicit constructions using the group of pure symmetric outer automorphisms. 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.