Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Let [inline-graphic not available: see fulltext] denote the field of algebraic numbers in [inline-graphic not available: see fulltext] A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ Md([inline-graphic not available: see fulltext](G, σ)), regarded as an operator on l2(G)d, the eigenvalues of A are algebraic numbers, where σ ∈ Z2(G, [inline-graphic not available: see fulltext]) is an algebraic multiplier, and [inline-graphic not available: see fulltext] denotes the unitary elements of [inline-graphic not available: see fulltext]. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σn=1 for some positive integer n) this property holds for a larger class of groups [inline-graphic not available: see fulltext] containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.