$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

Abstract: We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of Bader, Furman and Sauer. We apply this criterion to show that the categories constructed from totally disconnected groups by Arano and Vaes have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda<\Gamma$, with $\Gamma$ acting on a type $\mathrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes\Lambda\subset P\rtimes\Gamma$ to that of the Schlichting completion $G$ of $\Lambda<\Gamma$. If $\Lambda<\Gamma$ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimes\Lambda\subset P\rtimes\Gamma$ are equal to those of $G$.