L ²-betti numbers of rigid C*-tensor categories and discrete quantum groups

We compute the L ²-Betti numbers of the free C ^*-tensor categories, which are the representation categories of the universal unitary quantum groups A _u(F). We show that the L ²-Betti numbers of the dual of a compact quantum group G(double-struck) are equal to the L ²-Betti numbers of the representation category Rep(G(double-struck)) and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of L ²-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first L ²-Betti number in terms of a generating set of a C ^*-tensor category.