Discrete quantum subgroups of free unitary quantum groups

Abstract

We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_N^{+}$. In other words, we classify all discrete quantum subgroups of ${\hat{U}}_N^{+}$, thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.