Publication:
Existence of spectral gaps, covering manifolds and residually finite groups

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2008
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World Scientific Publishing
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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 Δ(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 Δ(X,gε) has, in addition, band-structure and there is an asymptotic estimate for the number N(λ) of components of spec Δ(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.
Description
33 pages, 2 figures.-- MSC1991 codes: 58J50, 35P15, 20E26, 57M10.-- Dedicated to Volker Enß on his 65th birthday.
MR#: MR2400010 (2008m:58072)
Zbl#: Zbl 1146.58021
Keywords
Covering manifolds, Spectral gaps, Residually finite groups, min-max principle
Bibliographic citation
Reviews in Mathematical Physics, 2008, vol. 20, n. 2, p. 199-231