#### Vol. 10, No. 7, 2017

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$L^2$-Betti numbers of rigid $C^*$-tensor categories and discrete quantum groups

### David Kyed, Sven Raum, Stefaan Vaes and Matthias Valvekens

Vol. 10 (2017), No. 7, 1757–1791
##### Abstract

We compute the ${L}^{2}$-Betti numbers of the free ${C}^{\ast }$-tensor categories, which are the representation categories of the universal unitary quantum groups ${A}_{u}\left(F\right)$. We show that the ${L}^{2}$-Betti numbers of the dual of a compact quantum group $\mathbb{G}$ are equal to the ${L}^{2}$-Betti numbers of the representation category $Rep\left(\mathbb{G}\right)$ 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 ${C}^{\ast }$-tensor category.

##### Keywords
$L^2$-Betti numbers, rigid $C^*$-tensor categories, discrete quantum groups, subfactors, compact quantum groups
##### Mathematical Subject Classification 2010
Primary: 46L37
Secondary: 16E40, 18D10, 20G42