A note on quantum subgroups of free quantum groups

In this short note, quantum subgroups in finite free products of the Pontryagin duals of free unitary quantum groups are classified. They correspond to pairs of a subgroup $\mathrm{\Gamma }$ and a subset $S$ of the free group ${\mathbb{F}}_n$ such that $S$ is $\mathrm{\Gamma }$-invariant, containing $\mathrm{\Gamma }$, and connected in the Cayley graph of ${\mathbb{F}}_n$