  Thoma's theorem states that a group algebra $C^*(\Gamma)$ is of type I if and
only if $\Gamma$ 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 $\Gamma$, we have a
stationary model of type $\pi:C^*(\Gamma)\to M_F(C(L))$, 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,
$\widehat{\Gamma}\subset S_N^+$, 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 $\leq1$.
