L2-Betti Numbers ofC*-Tensor Categories Associated with Totally Disconnected Groups

Abstract

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 $G$.

Robust Stereo Road Image Segmentation Using Threshold Selection Optimization Method Based on Persistent Homology

“…The homology group is an important topological invariant characteristic of a topological space, and the homology of a topological space can be measured by the Betti number [27]. Betti number is the rank of the homology group, and the nth dimensional Betti number is the number of n-dimensional holes on the topological space [28][29]. In this paper, we focus on the 0-dimensional Betti number, i.e., the number of connected components.…”