On Fenchel-Nielsen Coordinates of Surface Group Representations into SU(3,1)

Abstract

Let Σg be a compact, connected, orientable surface of genus g ≥ 2. We ask for a parametrisation of the discrete, faithful, totally loxodromic representations in the deformation space Hom(π1(Σg), SU(3, 1))/SU(3, 1). We show that such a representation, under some hypothesis, can be determined by 30g - 30 real parameters.