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

## TL;DR

This paper computes the $L^2$-Betti numbers for free $C^*$-tensor categories and discrete quantum groups, revealing invariance under monoidal equivalence and providing new calculations for quantum permutation and wreath product groups.

## Contribution

It establishes the invariance of $L^2$-Betti numbers under monoidal equivalence and computes these invariants for several classes of quantum groups and tensor categories.

## Key findings

- $L^2$-Betti numbers of dual quantum groups equal those of their representation categories
- New computations of $L^2$-Betti numbers for quantum permutation groups
- Upper bounds for the first $L^2$-Betti number based on generating sets

## Abstract

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 $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ 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.