Aut-invariant quasimorphisms on groups
Francesco Fournier-Facio, Richard D. Wade
TL;DR
The authors prove that a broad class of finitely generated groups admit infinite-dimensional spaces of Aut-invariant homogeneous quasimorphisms, including non-elementary hyperbolic, certain relatively hyperbolic, infinitely-ended groups, and many graph products not virtually abelian. Their approach blends the Bestvina–Fujiwara machinery with a pullback via the adjoint action from Aut(G) to Inn(G), leveraging acylindrical hyperbolicity whenever available. They further extend the construction to graph products and free products, yielding unbounded Aut-invariant norms and, in particular, unbounded stable autocommutator length on [G,G]. The paper also outlines open questions about extending these results to broader acylindrically hyperbolic groups and about the existence of Aut-invariant quasimorphisms in other contexts, including potential WWPD-based approaches and commensurability considerations.
Abstract
For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This was known for $F_2$ by a result of Brandenbursky and Marcinkowski, but is new even for free groups of higher rank, settling a question of Miklós Abért. The case of graph products of finitely generated abelian groups settles a question of Michal Marcinkowski. As a consequence, we deduce that a variety of Aut-invariant norms on such groups are unbounded.