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

Abstract

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 $\Lambda <\Gamma$, with $\Gamma$ acting on a type ${\mathrm{II}}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P⋊\Lambda \subset P⋊\Gamma$ 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 $P⋊\Lambda \subset P⋊\Gamma$ are equal to those of $G$.

44 pages, 1 figure. v2: Minor corrections. Added a remark and an example