Existence of spectral gaps, covering manifolds and residually finite groups

TL;DR

This paper demonstrates how to construct Riemannian coverings with prescribed spectral gaps in the Laplacian spectrum, using deformations and properties of residually finite groups, with implications for spectral theory and geometric analysis.

Contribution

It introduces two procedures for deforming coverings to produce spectra with spectral gaps and analyzes their spectral properties using group theory and existing results.

Findings

Constructed families of coverings with at least a prescribed number of spectral gaps.

Established band-structure and asymptotic estimates for the spectrum.

Provided examples of residually finite groups fitting the construction.

Abstract

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $\Delta_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $\Gamma$ has a positive Kadison constant, then we can apply results by Br\"uning and Sunada to deduce that $\spec \Delta_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,\lambda]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.