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

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