Arithmetic properties of eigenvalues of generalized Harper operators on graphs

Abstract

Let $\Qbar$ denote the field of complex algebraic numbers. A discrete group $G$ is said to have the $σ$-multiplier algebraic eigenvalue property, if for every matrix $A$ with entries in the twisted group ring over the complex algebraic numbers $M_d(\Qbar(G,σ))$, regarded as an operator on $l^2(G)^d$, the eigenvalues of $A$ are algebraic numbers, where $σ$ is an algebraic multiplier. 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 ($σ^n$=1 for some positive integer $n$) this property holds for a larger class of groups, 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.