  Let $X$ be a finite aspherical CW-complex whose fundamental group $\pi_1(X)$
possesses a subnormal series $\pi_1(X) \rhd G_m \rhd ... \rhd G_0$ with a
non-trivial elementary amenable group $G_0$. We investigate the
$L^2$-invariants of the universal covering of such a CW-complex $X$. We show
that the Novikov-Shubin invariants $\alpha_n({\tilde X})$ are positive. We
further prove that the $L^2$-torsion $\rho^{(2)}({\tilde X})$ vanishes if
$\pi_1(X)$ has semi-integral determinant.
