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Let $\Sigma_g$ be a compact, connected, orientable surface of genus $g \geq2$. We ask for a parametrization of the discrete, faithful, totally loxodromicrepresentations in the deformation space ${\rm Hom}(\pi_1(\Sigma_g), {\rmSU}(3,1))/{\rm SU}(3,1)$. We show that such a representation, under somehypothesis, can be determined by $30g-30$ real parameters.

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