The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps
type: Item
instance of: scholarly article
title: The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps
zbMATH Open document ID: 1236.11081
author: Yasuro Gon
author: Jinsung Park
published in: Mathematische Annalen
publication date: 2010-01-29T00:00:00Z
review text: In the article under review the Ruelle zeta-function and Selberg zeta-function, attached to the fundamental representations for real hyperbolic manifolds with cusps, are studied. The meromorphic extensions to all of \(\mathbb{C}\) and the functional equations are given. In addition, the order of the singularity of the Ruelle zeta-function at the origin is given. The key new step here is to completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations. This paper gives the proof in the case when the dimension of the manifold is even. The odd case was announced by the authors in [Proc. Japan Acad., Ser. A 84, No. 1, 1--4 (2008; Zbl 1204.11147)], and the proof can be found in [ \textit{U. Bunke} and \textit{M. Olbrich}, Selberg zeta and theta functions. A differential operator approach. Mathematical Research 83. Berlin: Akademie Verlag (1995; Zbl 0831.58001)], the first author [Math. Ann. 308, No. 2, 251--278 (1997; Zbl 0883.11022)], \textit{A. Juhl}, Cohomological theory of dynamical zeta functions, Progress in Mathematics 194. Basel: Birkhäuser (2001; Zbl 1099.37017)].
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5664551
zbMATH Keywords: Ruelle zeta-function
zbMATH Keywords: Selberg zeta-function
zbMATH Keywords: representations for real hyperbolic manifolds with cusps
zbMATH Keywords: meromorphic extensions
zbMATH Keywords: functional equations
zbMATH Keywords: weighted unipotent orbital integrals
zbMATH Keywords: singularity of Ruelle zeta-function at origin
zbMATH Keywords: Selberg trace formula
MaRDI profile type: Publication
full work available at URL: s00208-009-0408-7
OpenAlex ID: W1997348260
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DOI: 10.1007/S00208-009-0408-7
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description: scientific article; zbMATH DE number 5664551