$L^2$-Betti numbers of $C^*$-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 [BFS12]. We apply this criterion to show that the categories constructed from totally disconnected groups in [AV16] have vanishing L 2 -Betti numbers. Given an almost-normal inclusion of discrete groups Λ < Γ, with Γ acting on a type II 1 factor P by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion P Λ ⊂ P Γ 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 Λ ⊂ P Γ are equal to those of G.