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.
Foreman, Brendan, "Discrete Groups and the Complex Contact Geometry of SL(2, C)" (2010). Mathematics and Computer Science. 5.