Artin formalism for Selberg zeta functions of co-finite Kleinian groups
Abstract
Description
Let $Γ\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $Γ\subset {\rm SL}(2,\mathbb C)$ 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 $\tildeΓ$ is a finite index group extension of $Γ$ in ${\rm SL}(2,\mathbb C)$, and $π={\rm Ind}_Γ^{\tildeΓ}χ$ is the induced representation, then $Z(s;Γ;χ)=Z(s;\tildeΓ;π)$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $ϕ(s;Γ;χ)=ϕ(s;\tildeΓ;π)$, for an appropriate normalization of the Eisenstein series.
14 pages. In v2 added key reference and clarified relationship to certain results in the literature
14 pages. In v2 added key reference and clarified relationship to certain results in the literature