A quasimorphism (or quasicharacter) on a discrete group \(G\) is a function \(f : G \rightarrow \mathbb{R}\) such that the defect \(D(f)=\sup_{g,h \in G}\{ |f(g)+f(h)-f(gh) |\}\) is bounded. A quasimorphism is homogeneous (or a pseudocharacter) if it restricts to a homomorphism on every cyclic subgroup of \(G\). In the study of quasimorphisms, free groups have long been a central case study. This case is especially important, since homogeneous quasimorphisms on free groups can be used to compute stable commutator length on free groups, which in turn gives upper bounds on the stable commutator length of elements in any group (see [D. Calegari, slc. Tokyo: Mathematical Society of Japan. (2009; Zbl 1187.20035)]).
Given an automorphism \(\varphi \in \operatorname{Aut}(G)\) and a quasimorphism \(f : G \rightarrow \mathbb{R}\), we obtain a quasimorphism \(f \circ \varphi\). This yields an action of \(\operatorname{Aut}(G)\) on the space of quasimorphisms on \(G\), which preserves the subspace of homogeneous quasimorphisms. The main results proved in the paper under review are the following:
Theorem B: Let \(G\) be a finitely generated group and suppose that one of the following holds: (1) \(G\) is a non-elementary Gromov-hyperbolic group; (2) \(G\) is non-elementary hyperbolic relative to a collection of finitely generated subgroups \(\mathcal{P}\) and no group in \(\mathcal{P}\) is relatively hyperbolic; (3) \(G\) has infinitely many ends; (4) \(G\) is a graph product of finitely generated abelian groups on a finite graph and \(G\) is not virtually abelian. Then there exists an infinite-dimensional space of \(\operatorname{Aut}\)-invariant homogeneous quasimorphisms on \(G\).
All four items include the free group, which recovers the case of rank \(2\) [M. Brandenbursky and M. Marcinkowski, Comment. Math. Helv. 94, No. 4, 661–687 (2019; Zbl 1433.57008)]. Item (4) settles Problem 5.1 in [M. Marcinkowski, Mich. Math. J. 69, No. 2, 285–295 (2020; Zbl 1530.20107)] and recovers some results of [B. Karlhofer, “Aut-invariant quasimorphisms on graph products of abelian groups”, Preprint, arXiv:2107.12171]. Items (2) and (3) recovers the main result of [B. Karlhofer, Ann. Math. Qué. 47, No. 2, 475–493 (2023; Zbl 1535.20217)] for finitely generated groups.
Theorem E: Let \(G\) be a group and suppose that \(G\) is not virtually central and \(\operatorname{Aut}(G)\) is acylindrically hyperbolic. Then there exists an infinitedimensional space of \(\operatorname{Aut}\)-invariant homogeneous quasimorphisms on \(G\).