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 compact subgroup of G, and Γ a lattice in G. We study automorphic forms for Γ if G is of real rank one with some additional assumptions, using a dynamical approach based on properties of the homogeneous flow on Γ\G and a Livshitz type theorem we prove for such a flow. In the Hermitian case G = SU (n, 1) we construct relative Poincaré series associated to closed geodesies on Γ\G/K for one-dimensional representations of K, and prove that they span the corresponding spaces of holomorphic cusp forms.