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arXiv · 0801.1938

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

Eliot Brenner·Florin Spinu
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Abstract

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.

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Eliot Brenner, Florin Spinu. 2008-01-19. Artin formalism for Selberg zeta functions of co-finite Kleinian groups. https://arxiv.org/abs/0801.1938

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