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

Abstract

Let ${\Sigma }_g$ be a compact, connected, orientable surface of genus $g\ge 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space $\mathrm{Hom}({\pi }_1({\Sigma }_g),\mathrm{SU}(3,1))/\mathrm{SU}(3,1)$. We show that such a representation, under some hypothesis, can be determined by $30g-30$ real parameters.

arXiv commentCorrected typos, updated bibliography