Aut-Invariant Word Norm on Right-Angled Artin and Right-Angled Coxeter Groups

Abstract

We prove that the $\mathrm{Aut}$-invariant word norm on right-angled Artin and right-angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness, we exhibit certain characteristic subgroups. This allows us to find unbounded quasi-morphisms which are Lipschitz with respect to the $\mathrm{Aut}$-invariant word norm.