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.
Linear programming and the intersection of free subgroups in free products of groups
The Space of Traces of the Free Group and Free Products of Matrix Algebras
On the cyclic subgroup separability of free products of two groups with amalgamated subgroup
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