A note on the Selberg zeta function for compact quotients of hyperbolic spaces
type: Item
instance of: scholarly article
title: A note on the Selberg zeta function for compact quotients of hyperbolic spaces
zbMATH Open document ID: 0829.22019
author: Masato Wakayama
published in: Hiroshima Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(G\) be a connected component of the isometric transformation group of \(\widetilde X\). Let \(K\) be a maximal compact subgroup of \(G\). Then \(\widetilde X\) is realized by \(G/K\), a noncompact symmetric space of rank one. Let \(\Gamma\) be a discrete torsion-free subgroup of \(G\) such that \(\Gamma \setminus G\) is compact. Thus our manifold \(X\) considered throughout this note is given by \(\Gamma \setminus G/K\) for some such \(\Gamma\). Let \(T\) be a finite-dimensional unitary representation of \(\Gamma\), and let \(\chi\) be its character. In the monumental work [J. Indian Math. Soc., n. Ser. 20, 47-87 (1956; Zbl 0072.082)], \textit{A. Selberg} constructed a function of complex variable \(Z_\Gamma (s, \chi)\), which is called Selberg's zeta function, and showed how the location and the order of its zeroes gave information about the spectrum of the Laplacian for \(X\) on the one hand and about the topology of \(X\) on the other. Furthermore, in the famous paper [Illinois J. Math. 21, 1-41 (1977; Zbl 0354.33013)] \textit{R. Gangolli} extended these results to a more general situation, that is to say, to the present case. In a previous paper [J. Math. Soc. Japan 41, 463-471 (1989; Zbl 0684.10039)] we have worked out a new method of the analytic continuation of the logarithmic derivative of Selberg's zeta function for compact Riemannian surfaces \(X\). In the present note, we consider the generalization of these results when \(X\) is a certain compact quotient of a hyperbolic space \(\widetilde {X}\).
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 63001
zbMATH Keywords: compact quotient of hyperbolic space
zbMATH Keywords: Selberg's zeta function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206128720
OpenAlex ID: W1533443505
Wikidata QID: Q128873166
DOI: 10.32917/HMJ/1206128720
description: scientific article; zbMATH DE number 63001