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 $\Gamma \subset \mathrm{SL}(2,\mathbb{C})$ 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 $\mathrm{SL}(2,\mathbb{C})$, and $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ 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 $\varphi (s;\Gamma ;\chi )=\varphi (s;\tilde{\Gamma };\pi )$, for an appropriate normalization of the Eisenstein series.

14 pages. In v2 added key reference and clarified relationship to certain results in the literature