DISCRETE GROUPS AND THE COMPLEX CONTACT GEOMETRY OF Sl(2,C)

We investigate the vertical foliation of the standard complex contact structure on Γ \ Sl(2,C), where Γ is a discrete subgroup. We find that, if Γ is nonelementary, the vertical leaves on Γ \Sl(2,C) are holomorphic but not regular. However, if Γ is Kleinian, then Γ \ Sl(2,C) contains an open, dense set on which the vertical leaves are regular, complete and biholomorphic to C∗. If Γ is a uniform lattice, the foliation is nowhere regular, although there are both infinitely many compact and infinitely many nonclosed leaves.