Existence of spectral gaps, covering manifolds and residually finite groups

Fernando Lledó, Olaf Post
March 03, 2005
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 $(X,g_\varepsilon) \to (M,g_\varepsilon)$ such that the spectrum of the Laplacian $\Delta_{(X_\varepsilon,g_\varepsilon)}$ has at least a prescribed finite number of spectral gaps provided $\varepsilon$ is small enough. If $\Gamma$ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec$\Delta_{(X,g_\varepsilon)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of spec${\Delta {(X,g_\varepsilon)}}$ 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.