Existence of spectral gaps, covering manifolds and residually finite groups

Existence of spectral gaps, covering manifolds and residually finite groups (2007)

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by Fernando Lledó , Olaf Post

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@MISC{Lledó07existenceof,
    author = {Fernando Lledó and Olaf Post},
    title = {Existence of spectral gaps, covering manifolds and residually finite groups},
    year = {2007}
}

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Abstract

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, gε) → (M, gε) 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 ∆ (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.

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