On the bottom of spectra under coverings

Abstract

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.

arXiv comment8 pages, fixed a technical mistake concerning the volume of the boundary of fundamental domains