Bottom of spectra and amenability of coverings

TL;DR

This paper investigates the relationship between the bottom of the spectra of Riemannian coverings and their amenability, establishing conditions under which spectral equality implies amenability.

Contribution

It proves that under certain geometric conditions, the equality of spectral bottoms implies the covering is amenable, extending previous results.

Findings

Spectral bottoms coincide for amenable coverings.

The converse holds under completeness and Ricci curvature bounds.

A natural spectral condition ensures the converse implication.

Abstract

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 $M_0$.