Complex hyperbolic Fenchel–Nielsen coordinates

Abstract

Let $\Sigma$ be a closed, orientable surface of genus $g$. It is known that the $\mathrm{SU}(2,1)$ representation variety of ${\pi }_1\left(\Sigma \right)$ has $2g-3$ components of (real) dimension $16g-16$ and two components of dimension $8g-6$. 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 $\Sigma$ and complex Fenchel–Nielsen coordinates on the (classical) quasi-Fuchsian space of $\Sigma$.