𝐴𝑢𝑡-invariant quasimorphisms on groups

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 F_2 by a result of Brandenbursky and Marcinkowski [Comment. Math. Helv. 94 (2019), pp. 661–687], 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 Michał Marcinkowski. As a consequence, we deduce that a variety of A u t Aut -invariant norms on such groups are unbounded.