Interplay Between the Algebraic Structure of a Group and Arithmetic Properties of its Spectrum

We investigate the inteplay between the algebraic structure of a group G and arithmetic properties of its spectrum σ(G) which consists of the eigenvalues of all the inner automorphisms of G. A complex number λ is called an eigenvalue of a group automorphism A : G → G if φ ◦ A|H = λ · φ for some non-trivial homomorphism φ : H → C defined on an A-invariant subgroup H ⊂ G. It is shown that many properties of a group G (such as the presence of a finitely generated subgroup of infinite rank, nilpotence, periodicity, polycyclicity, etc) are coded in its spectrum. In the paper the spectra are applied to investigate the so-called reversive properties of groups. The paper ends with a list of related open problems.