A note on the Selberg zeta function for compact quotients of hyperbolic spaces
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 \(\widetilde {X}\).
scientific article; zbMATH DE number 63001