EXISTENCE OF SPECTRAL GAPS, COVERING MANIFOLDS AND RESIDUALLY FINITE GROUPS

FERNANDO LLEDÓ, OLAF POST
2008 Reviews in Mathematical Physics  
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 ε ) → (M, g ε ) such that the spectrum of the Laplacian ∆ (Xε,gε) has at least a prescribed finite number of spectral gaps provided ε is small enough. If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec ∆
more » ... uce that spec ∆ (X,gε) has, in addition, band-structure and there is an asymptotic estimate for the number N (λ) of components of spec ∆ (X,gε) 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.
doi:10.1142/s0129055x08003286 fatcat:iaxy6cusyfauze64xwsagngqtm