Selberg zeta functions for spaces of higher rank

5 Introduction In 1956 A. Selberg introduced the zeta function Z(s) = c N ≥0 (1 − e −(s+N)l(c)), Re(s) >> 0, where the first product is taken over all primitive closed geodesics in a compact Riemannian surface of genus ≥ 2, equipped with the hyperbolic metric, and l(c) denotes the length of the geodesic c. Selberg proved that the product converges if the real part of s is large enough and that Z(s) extends to an entire function on C, that it satisfies a functional equation as s is replaced by 1 − s, and that a generalized Riemann hypothesis holds for the function Z. The following list of references is only a small and strongly subjective selec-Generalizing Selberg's definition R. Gangolli [32] gave a zeta function with similar properties for all locally symmetric spaces of rank one. These results were later extended to vector bundles by M. Wakayama [74]. Both, Gangolli and Wakayama kept the spirit of Selberg's original proof in that they explicitly inverted the Selberg integral transform (which required more knowledge on harmonic analysis then in Selberg's case, however), then handicrafted a test function to be plugged into the trace formula which would yield a higher logarithmic derivative of the Selberg zeta function. In [31] D. Fried introduced heat kernel methods which replaced the inversion of the transform conveniently. Using supersymmetry arguments his approach was transferred to some cases of higher rank by H. Moscovici and R. Stanton [63]. In the present paper a new approach is given which allows to extend the theory of the Selberg zeta function to arbitrary groups. The central idea is to use a geometric approach to shift down to a Levi subgroup of rank one less and then plug in Euler-Poincaré functions with respect to that Levi group. The themes of the paper are: meromorphicity of the Selberg zeta function, the Patterson conjecture relating the vanishing order of the zeta function to group cohomology, and the evaluation of the zeta function at special points to get topological invariants. The contents of the chapters are as follows: In the first chapter notations are set and results from literature are collected. In chapter 2 the existence 6 of pseudo-coefficients and Euler-Poincaré functions is proved in a very general setting. Euler-Poincaré functions are characterized by the fact that they give Euler-Poincaré characteristics as traces under irreducible representations. They are of great use in …