Aut-invariant quasimorphisms on free products

TL;DR

This paper constructs automorphism-invariant quasimorphisms on free products of groups, showing their abundance and applying this to demonstrate non-triviality of an invariant stable commutator length.

Contribution

It explicitly constructs automorphism-invariant quasimorphisms on free products and proves their infinite-dimensionality for most cases, advancing understanding of group invariants.

Findings

Invariant quasimorphisms form an infinite-dimensional space for most free products.

Invariant stable commutator length is non-trivial on these groups.

Explicit construction of automorphism-invariant quasimorphisms.

Abstract

Let $G=A \ast B$ be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on $G$ which are invariant with respect to all automorphisms of $G$. We also prove that the space of such quasimorphisms is infinite-dimensional whenever $G$ is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.