- 05-245 Lled\'o, Fernando and Post, Olaf
- Existence of spectral gaps, covering manifolds and residually finite groups
Jul 15, 05
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Abstract. In this paper we present two construction procedures of covering manifolds $X \to M$ with residually finite covering transformation group $\Gamma$ such that the spectrum of the Laplacian $\Delta_X$ has at least a prescribed number $n$ of spectral gaps, $n \in \N$. If $\Gamma$ has a positive Kadison constant, then we can apply results by Br\"uning and Sunada to the manifolds constructed here. In this case spec $\Delta_X$ has, in addition, band structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of
spec $\Delta_X$ that intersect the interval $[0,\lambda]$. Finally, we present several classes of examples of residually finite groups that fit with our construction procedure and study their mutual relations.