$L^2$-Betti numbers of $C^{\ast }$-tensor categories associated with totally disconnected groups

We prove that the $L^2$-Betti numbers of a rigid $C^{\ast }$-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 $\mathrm{\Lambda }<\mathrm{\Gamma }$, with $\mathrm{\Gamma }$ acting on a type ${\mathrm{I}\mathrm{I}}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes\Lambda\subset P\rtimes\Gamma$tothatoftheSchlichtingcompletion$G$of$\Lambda<\Gamma$. If $\mathrm{\Lambda }<\mathrm{\Gamma }$ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P⋊\mathrm{\Lambda }\subset P⋊\mathrm{\Gamma }$ are equal to those of $G$.