  For a Riemannian covering $\pi\colon M_1\to M_0$, the bottoms of the spectra
of $M_0$ and $M_1$ coincide if the covering is amenable. The converse
implication does not always hold. Assuming completeness and a lower bound on
the Ricci curvature, we obtain a converse under a natural condition on the
spectrum of $M_0$.
