Discrete groups and the complex contact geometry of 𝑆𝑙(2,ℂ)

Abstract

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