Complex hyperbolic Fenchel–Nielsen coordinates

Let ΣΣ be a closed, orientable surface of genus gg. It is known that the SU(2,1) representation variety of π1(Σ)π1(Σ) has 2g−32g−3 components of (real) dimension 16g−1616g−16 and two components of dimension 8g−68g−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 ΣΣ and complex Fenchel–Nielsen coordinates on the (classical) quasi-Fuchsian space of ΣΣ.