The space of quasi-Fuchsian surface group representations into PSL(2,C) has been largely studied in recent years, also due to the direct relation with hyperbolic geometry. In this framework, Bers' simultaneous uniformization theorem shows that it is parameterized by two copies of Teichmüller space of the closed surface. In this seminar, we are interested in surface group representations into SL(3,C) acting naturally on a homogeneous space called the bi-complex hyperbolic space. After briefly describing its main features, we will define minimal complex Lagrangian surface immersions and their structural equations. If they are equivariant under the action of a representation into SL(3,C), we will see that their embedding data provide a parameterization of the space of SL(3,C)-quasi-Fuchsian representations by two copies of the bundle of holomorphic cubic differentials over Teichmüller space. This is a joint work with A. Tamburelli.
