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

Abstract:

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.