Lück W. and Schick, T.: Approximating L2-signatures by their finite dimensional analogues
Abstract: Let $\Gamma$ be a group together with a sequence
of normal subgroups $\Gamma\supset \Gamma_1\supset \Gamma_2\supset\dots$
of finite index $[\Gamma:\Gamma_k]$ such
that $\bigcap_k \Gamma_k=\{1\}$.
Let $(X,Y)$ be a (compact) $4n$-dimensional Poincar\'e
pair and $p: (\overline{X},\overline{Y}) \to (X,Y)$ be a
$\Gamma$-covering, i.e. normal
covering with $\Gamma$ as deck transformation group.
We get associated $\Gamma/\Gamma_k$-coverings $(X_k,Y_k) \to (X,Y)$.
We prove that
\begin{equation*}
\sign^{(2)}(\overline{X},\overline{Y}) = \lim_{k\to\infty}
\frac{\sign(X_k,Y_k)}{[\Gamma : \Gamma_k]},
\end{equation*}
where $\sign$ or $\sign^{(2)}$
is the signature or $L^2$-signature, respectively, and the
convergence of the right side for
any such sequence $(\Gamma_k)_{k \ge 1}$ is part
of the statement.
If $\Gamma$ is amenable, we prove a similar
approximation theorem for $\sign^{(2)}(\overline{X},\overline Y)$ in terms of the
signatures of a regular exhaustion of $\overline{X}$.
Thomas Schick