Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups Articles uri icon

publication date

  • March 2008

start page

  • 199

end page

  • 231

issue

  • 2

volume

  • 20

International Standard Serial Number (ISSN)

  • 0129-055X

abstract

  • In the present paper we consider Riemannian coverings (X, g) → (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, gepsilon) → (M, gepsilon) such that the spectrum of the Laplacian Delta(Xepsilon,gepsilon) has at least a prescribed finite number of spectral gaps provided epsilon is small enough. If Gamma has a positive Kadison constant, then we can apply results by BrÅ¡uning and Sunada to deduce that spec Delta(X,gepsilon) has, in addition, band-structure and there is an asymptotic estimate for the number N(lambda) of components of spec Delta(X,gepsilon) 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.