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

TL;DR

This paper proves an Artin formalism for Selberg zeta functions and scattering functions of co-finite Kleinian groups, showing their invariance under finite index group extensions with induced representations.

Contribution

It establishes the Artin formalism for Selberg zeta functions and scattering functions in the context of co-finite Kleinian groups, extending previous results to this setting.

Findings

Proves $Z(s;\Gamma;\chi)=Z(s; ilde\Gamma;\pi)$ for induced representations.

Establishes $\phi(s;\Gamma;\chi)=\phi(s; ilde\Gamma;\pi)$ for scattering functions.

Validates the formalism for both zeta and scattering functions in this geometric setting.

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.