Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces

Let G be a semisimple Lie group with no compact factors, K a maximal compac t subgroup of G, and Gamma a lattice in G. We study automorphic forms for G amma if G is of real rank one with some additional assumptions, using a dyn amical approach based on properties of the homogeneous flow on Gamma \G and a Livshitz type theorem we prove for such a flow. In the Hermitian case G = SU(n, 1) we construct relative Poincare series associated to closed geode sics on Gamma \G/K for one-dimensional representations of K, and prove that they span the corresponding spaces of holomorphic cusp forms.