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 [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 II1 factor P by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion P x Lambda subset of P x 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 x Lambda subset of P x Gamma are equal to those of G.

Automated summary

This paper proves that the L-2-Betti numbers of a rigid C*-tensor category vanish when there is an almost-normal subcategory with vanishing L-2-Betti numbers, extending a result from [7]. The criterion is applied to show that the categories constructed from totally disconnected groups in [6] have vanishing L-2-Betti numbers. Additionally, the paper relates the cohomology theory of the quasi-regular inclusion P x Lambda subset of P x Gamma to that of the Schlichting completion G of Lambda < Gamma in the case of an almost-normal inclusion of discrete groups Lambda < Gamma, with Gamma acting on a type II1 factor P by outer automorphisms.