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

Abstract

Let $\Gamma\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $\Gamma\subset {\rm SL}(2,\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one associates the Selberg zeta function $Z(s;\Gamma;\chi)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde\Gamma$ is a finite index group extension of $\Gamma$ in ${\rm SL}(2,\mathbb C)$, and $\pi={\rm Ind}_{\Gamma}^{\tilde\Gamma}\chi$ is the induced representation, then $Z(s;\Gamma;\chi)=Z(s;\tilde\Gamma;\pi)$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi(s;\Gamma;\chi)=\phi(s;\tilde\Gamma;\pi)$, for an appropriate normalization of the Eisenstein series.