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

Let ${\mathrm{\Sigma }}_g$ be a compact, connected, orientable surface of genus $g \geq 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space ${\rm Hom}(\pi_1(\Sigma_g), {\rm SU}(3,1))/{\rm SU}(3,1)$. We show that such a representation, under some hypothesis, can be determined by $30g-30$ real parameters.