Thoma type results for discrete quantum groups

Thoma's theorem states that a group algebra C∗(γ) is of type I if and only if γ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group γ, we have a stationary model of type π: C∗(γ) → M F(C(L)), with F being a finite quantum group, and with L being a compact group. We discuss then some refinements of these results in the quantum permutation group case, π C SN+, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank ≤ 1.