Traces in complex hyperbolic geometry

A complex hyperbolic orbifold M can be written as HC/Γ where Γ is a discrete, faithful representation of π1(M) to Isom(H 2 C). The group SU(2, 1) is a triple cover of the group of holomorphic isometries of HC and (taking subgroups if necessary) we view Γ as a subgroup of SU(2, 1). Our main goal is to discuss the connection between the geometry of M and traces of Γ. We do this in two specific cases. First, we consider the case where Γ is a free group on two generators, which we view as the fundamental group of a three holed sphere. We indicate how to use this analysis to give Fenchel-Nielsen coordinates on the complex hyperbolic quasi-Fuchsian space of a surface of genus g ≥ 2. Secondly, we consider the case where Γ is a triangle group generated by complex reflections in three complex lines. We keep in mind similar results from the more familiar setting of Fuchsian and Kleinian groups and we explain those examples from our point of view.