Artin formalism for Selberg zeta functions of co-finite Kleinian groups

الأصل · EN

Let Γ³ be a finite-volume quotient of the upper-half space, where Γ⊂ SL(2,C) is a discrete subgroup. To a finite dimensional unitary representation χ of Γ one associates the Selberg zeta function Z(s;Γ;χ). In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if Γ is a finite index group extension of Γ in SL(2,C), and π= IndΓΓχ is the induced representation, then Z(s;Γ;χ)=Z(s;Γ;π). In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely ϕ(s;Γ;χ)=ϕ(s;Γ;π), for an appropriate normalization of the Eisenstein series.