The Selberg zeta function and a local trace formula for Kleinian groups. (English) Zbl 0697.10027
Let \(\Gamma\) be a geometrically finite discrete group of isometries of real hyperbolic n-dimensional space \({\mathbb{H}}^ n\). We suppose that \(\Gamma\) is convex co-compact, contains no elliptic elements, and that the quotient \(M={\mathbb{H}}^ n/\Gamma\) has infinite hyperbolic volume. We study the Selberg zeta function introduced by Patterson for this class of groups and prove a trace formula refining and generalizing earlier results of Patterson. This trace formula connects the poles of the scattering operator to those of the meromorphically continued logarithmic derivative of the zeta function.
Reviewer: P.A.Perry
MSC:
11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
11M35 | Hurwitz and Lerch zeta functions |