L2-Betti numbers of rigid C∗-tensor categoriesand discrete quantum groups

Abstract

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which\nare the representation categories of the universal unitary quantum groups\n$A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum\ngroup $G$ are equal to the $L^2$-Betti numbers of the representation category\n$Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an\napplication, we obtain several new computations of $L^2$-Betti numbers for\ndiscrete quantum groups, including the quantum permutation groups and the free\nwreath product groups. Finally, we obtain upper bounds for the first\n$L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.\n