Aut-invariant quasimorphisms on groups

TL;DR

This paper constructs infinite-dimensional spaces of automorphism-invariant quasimorphisms for various complex groups, revealing new properties and unbounded norms, and settling several open questions in group theory.

Contribution

It introduces new automorphism-invariant quasimorphisms for broad classes of groups, including hyperbolic and graph product groups, extending known results and resolving open problems.

Findings

Existence of infinite-dimensional automorphism-invariant quasimorphisms in many groups

Unbounded Aut-invariant norms on these groups

Resolution of questions for free groups and graph products

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 Mikl\'os Ab\'ert. 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.