Existence of spectral gaps, covering manifolds and residually finite groups

Abstract

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $Γ$ 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 $Δ_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $Γ$ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that $\spec Δ_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(λ)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,λ]$. 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.

final version (26 pages, 2 figures). to appear in Rev. Math. Phys