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

Abstract

Let Γ\H 3 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 = Ind Γ Γ χ 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.

Reduced 2-Dimensional Birkhoff Surfaces of Section of the Desymmetrized P Sl(2, Z) Group: The Anosov Characterization

“…The Artin formalism of the desymmetrized PSL(2,Z) group is applied in [3]. Particular cases of the Artin formalism provided with the Selberg zeta function are exposed in [4].…”