Approximating L^2-signatures by their compact analogues

Abstract

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement.