Closed geodesics in compact nilmanifolds

We study the density of closed geodesies property on 2-step nilmanifolds Γ \ N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra n-fraktur sign, and Γ is a lattice in N. We show the density of closed geodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU (2).