Approximating L 2-Signatures by Their Compact Analogues

Abstract

Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ 1 ⊃ Γ 2 ... of finite index [Γ : Γ k ] such that ⋂ k Γ k = {1}. Let ( X , Y ) be a (compact) 4 n -dimensional Poincaré pair and p : ( X ̄, Y ̄) → ( X , Y ) be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γ k -coverings ( X k , Y k ). We prove that where sign or sign (2) is the signature or L 2 -signature, respectively, and the convergence of the right side for any such sequence (Γ k ) k ≥1 is part of the statement. If Γ is amenable, we prove in a similar way an approximation theorem for sign (2) ( X ̄, Y ̄) in terms of the signatures of a regular exhaustion of X ̄. Our results are extensions of Lück's approximation results for L 2 -Betti numbers [10, Theorem 0.1].