Existence of spectral gaps, covering manifolds and residually finite groups

Original · EN

In the present paper we consider Riemannian coverings (X,g) → (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) → (M,g) such that the spectrum of the Laplacian Δ(X,g) has at least a prescribed finite number of spectral gaps provided is small enough. If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that Δ(X,g) has, in addition, band-structure and there is an asymptotic estimate for the number N(λ) of components of (X,g) 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.