Some negatively curved manifolds with cusps, mixing and counting

Let X be a Hadamard manifold whose sectional curvature K satisfies -b(2) less than or equal to K less than or equal to -1. We consider a family of free isometry groups Gamma acting properly discontinuously on X and containing parabolic transformations of divergence type. We show that such groups are of divergent type, we describe the dynamic properties of the map T induced by the action of Gamma on the boundary of X and we explore the spectrum of the transfer operator associated with T. As applications, we establish a mixing property for the geodesic flow on the unit tangent bundle of X/Gamma and we describe the behaviour as a goes to +infinity of the number of primitive closed geodesics on X/Gamma whose length is not larger than a.