Discrete groups and the complex contact geometry of $Sl(2,\mathbb{C})$

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