Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups

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

Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups Articles