On the structure of quantum automorphism groups

We compute the $K$-theory of quantum automorphism groups of finite dimensional $C^{\ast }$-algebras in the sense of Wang. The results show in particular that the $C^{\ast }$-algebras of functions on the quantum permutation groups $S_n^{+}$ are pairwise non-isomorphic for different values of $n$. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena.