An analytic characterization of freeness for finitely generated discrete quantum groups

We prove that a freer quantum group has smaller moments of the self-adjoint main character in the category of finitely generated discrete quantum groups. As a result, the moments are minimized precisely by the unitary free quantum groups $\mathbb{F}U(Q)$. Furthermore, in the spirit of [CC22], we prove that the operator norm of the self-adjoint main character is minimized only by unitary free quantum groups, at least in the subcategory of duals of free quantum groups of Kac type.