  For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with
boundary (possibly empty) and respective fundamental groups
$\Gamma_1\subseteq\Gamma_0$, we show that the bottoms of the spectra of $M_0$
and $M_1$ coincide if the right action of $\Gamma_0$ on
$\Gamma_1\backslash\Gamma_0$ is amenable.
