A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups

We present a link between easy quantum groups, discrete groups and combinatorics. By this, we infer new connections between quantum isometry groups, reflection groups, varieties of groups and the combinatorics of partitions. More precisely, we consider easy quantum groups and find a relation to subgroups of the infinite free product ${\mathbb{Z}}_2^{\ast \mathrm{\infty }}$ of $\mathbb Z_2=\mathbb Z/2\mathbb Z$. We obtain a link with reflection groups and thus with varieties of groups, which yields a statement on the complexity of the class of easy quantum groups on the one hand, and a “quantum invariant” for varieties of groups on the other hand. Moreover, we reveal a triangular relationship between easy quantum groups, categories of partitions and discrete groups (reflection groups). As a by-product, we obtain a large number of new quantum isometry groups.