论文标题
$ l^2 $ -betti数字的$ C^*$ - 张量类别与完全断开的组相关
$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups
论文作者
论文摘要
我们证明，$ l^2 $ -betti刚性$ c^*$ - 张量类别在存在几乎正常的子类别的情况下消失，消失了$ l^2 $ -Betti数字，这是Bader，Furman和Sauer的结果。我们应用此标准以表明Arano和VAE完全断开的组构建的类别消失了$ L^2 $ -BETTI数字。考虑到离散$λ<γ$的几乎正常的包含，$γ$作用于外部自动形态的类型$ \ mathrm {ii} _1 $ proact $ p $ p $，我们将Quasi-rativusion $ p \ rtimespimespimespimesp \ subset p \ subset p \ rtimes $ rtimes $ $ $ $ g的共同体理论联系起来。如果$λ<γ$是单模型的，则此通信使我们能够证明$ l^2 $ -betti $ p \rtimesλ\ subset p \rtimesγ$等于$ g $。
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 $G$.
