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

2008/01/13 by Eliot Brenner, Brenner, Eliot, Florin Spinu +1
Mathematics · #11F72 #11M36 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #math.NT #msc:11F72 #msc:11M36

paper · pdf · doi:10.48550/arxiv.0801.1938

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

openalex publication_date 2008/01/13 · arxiv created 2008/01/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ\backslash\mathbb H³ be a finite-volume quotient of the upper-half space, where Γ⊂ \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 Γ is a finite index group extension of Γ in \rm SL(2,\mathbb C), and π=\rm Ind_Γ^Γχ is the induced representation, then Z(s;Γ;χ)=Z(s;Γ;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;Γ;π), for an appropriate normalization of the Eisenstein series.

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