The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps
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)].
representations for real hyperbolic manifolds with cusps
singularity of Ruelle zeta-function at origin
scientific article; zbMATH DE number 5664551
scientific article