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

Abstract

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 dynamical approach based on properties of the homogeneous flow on $Γ\backslash G$ 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 geodesics on $Γ\backslash G/K$ for one-dimensional representations of K, and prove that they span the corresponding spaces of cusp forms.