Approximating L-signatures by their compact analogues

Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ1 ⊃ Γ2 ⊃ . . . of finite index [Γ : Γk] such that T k Γk = {1}. Let (X,Y ) be a (compact) 4n-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/Γk-coverings (Xk, Yk)→ (X,Y ). We prove that sign(X,Y ) = lim k→∞ sign(Xk, Yk) [Γ : Γk] , where sign or sign is the signature or L-signature, respectively, and the convergence of the right side for any such sequence (Γk)k≥1 is part of the statement. If Γ is amenable, we prove in a similar way an approximation theorem for sign(X,Y ) in terms of the signatures of a regular exhaustion of X. Our results are extensions of Lück’s approximation results for L-Betti numbers [10, Theorem 0.1].