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

TL;DR

This paper investigates a parametrization of certain discrete, faithful representations of surface groups into SU(3,1), showing they can be described by 30g-30 real parameters under specific conditions.

Contribution

It introduces a new parametrization of surface group representations into SU(3,1) using Fenchel-Nielsen coordinates, quantifying them with 30g-30 parameters.

Findings

Representations can be parametrized by 30g-30 real parameters.

Under certain hypotheses, the parametrization is complete.

Provides a framework for understanding surface group representations into SU(3,1).

Abstract

Let $\Sigma_g$ 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.