# On the bottom of spectra under coverings

## TL;DR

This paper proves that under certain conditions involving group actions, the lowest spectral values of a Riemannian manifold and its covering are equal, extending understanding of spectral properties in geometric analysis.

## Contribution

It establishes a new link between the amenability of group actions and the equality of spectral bottoms in Riemannian coverings.

## Key findings

- Spectral bottoms coincide under amenable group actions
- Amenability of the group action is key to spectral equality
- Extends spectral theory in Riemannian geometry

## 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.