Complex lines in complex hyperbolic space H ²_ℂ

Abstract

Let p ₁, p ₂, p ₃, p ₄ be four pairwise distinct points in the boundary of complex hyperbolic 2-space H ²_ℂ and any three points do not lie in the same C-circle. We show that we are always able to group the four points into two classes such that each class contains two points, the two complex lines spanned by each class are ultra-parallel or intersect. As an application, we can simplify the discussion in the paper [7], in which Parker and Platis used the global geometry coordinates to describe the Falbel’s cross-ratio variety of the four pairwise distinct points on the ∂H ²_ℂ .