Complex hyperbolic Fenchel–Nielsen coordinates

Let Σ be a closed, orientable surface of genus g . It is known that the SU ( 2 , 1 ) representation variety of π 1 ( Σ ) has 2 g − 3 components of (real) dimension 16 g − 16 and two components of dimension 8 g − 6 . Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety that contains the quasi-Fuchsian representations. These coordinates are a natural generalisation of Fenchel–Nielsen coordinates on the Teichmüller space of Σ and complex Fenchel–Nielsen coordinates on the (classical) quasi-Fuchsian space of Σ .