  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 $\mathbb Z_2^{*\infty}$ 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.
