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 [ 7]. We apply this criterion to show that the categories constructed from totally disconnected groups in [ 6] have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda <\Gamma $, with $\Gamma $ acting on a type $\textrm{II}_1$ 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 $G$.

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L 2-Betti 数的 C*-Tensor 类别与完全不连接的组相关联

我们证明了刚性 $C^*$-张量类别的 $L^2$-Betti 数在存在具有消失的 $L^2$-Betti 数的几乎正态子类别的情况下消失，概括了 [7 ]。我们应用这个标准来表明 [6] 中由完全不连贯的组构成的类别具有消失的 $L^2$-Betti 数。给定离散群 $\Lambda <\Gamma $ 的几乎正态包含，其中 $\Gamma $ 通过外部自同构作用于类型 $\textrm{II}_1$ 因子 $P$，我们将上同调理论联系起来准正则包含 $P\rtimes \Lambda \subset P\rtimes \Gamma $ 到 $\Lambda <\Gamma $ 的 Schlichting 完成 $G$ 的包含。如果$\Lambda <\Gamma $ 是单模的，这个对应关系可以证明$P\rtimes \Lambda \subset P\rtimes \Gamma $ 的$L^2$-Betti 数等于$G美元。