Functiones et Approximatio Commentarii Mathematici

Loxodromic Eisenstein series for cofinite Kleinian groups

Yosuke Irie

Advance publication

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Abstract

We introduce an Eisenstein series associated to a loxodromic element of cofinite Kleinian groups, namely the loxodromic Eisenstein series, and study its fundamental properties. It is the analogue of the hyperbolic Eisenstein series for Fuchsian groups of the first kind. We prove the convergence and the differential equation associated to the Laplace-Beltrami operator. We also prove the precise spectral expansion associated to the Laplace-Beltrami operator. Furthermore, we derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.

Article information

Source
Funct. Approx. Comment. Math., Advance publication (2018), 17 pages.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1543460444

Digital Object Identifier
doi:10.7169/facm/1781

Subjects
Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

Keywords
loxodromic Eisenstein series Kleinian group

Citation

Irie, Yosuke. Loxodromic Eisenstein series for cofinite Kleinian groups. Funct. Approx. Comment. Math., advance publication, 29 November 2018. doi:10.7169/facm/1781. https://projecteuclid.org/euclid.facm/1543460444


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