Complex hyperbolic Fenchel-Nielsen coordinates.

Abstract

Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π1(Σ) 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 Σ and complex Fenchel–Nielsen coordinates on the (classical) quasi-Fuchsian space of Σ.