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

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 $Λ<Γ$, with $Γ$ acting on a type $\mathrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimesΛ\subset P\rtimesΓ$ to that of the Schlichting completion $G$ of $Λ<Γ$. If $Λ<Γ$ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimesΛ\subset P\rtimesΓ$ are equal to those of $G$.