Poincar E Series and Comparison Theorems for Variable Negative Curvature

Poincaré series have always played a central rôle in the theory of automorphic functions and harmonic analysis on manifolds of constant negative curvature. The trace formulae approach was introduced by Selberg [20] in 1956 and has become the key to their analysis. However, with the advent of a more geometrical viewpoint (through the work of Patterson [15, 16], Sullivan [21, 22] and others) it becomes natural to broaden the scope to Poincaré series for manifolds of variable negative curvature. The purpose of this note is to describe an ergodic theoretic method for analyzing Poincaré series in this greater generality. Let M be a compact Riemannian manifold (possibly with totally geodesic boundary) with strictly negative variable curvatures. Let π1(M) denote its fundamental group. For any point p ∈M we can associate to each homotopy class γ ∈ π1(M)−{e} the length l(γ) of the shortest geodesic arc in γ from p to itself. We define the associated Poincaré series by η(s) = ∑