Local Rigidity for PGL(3, C)-Representations of 3-Manifold Groups

Let M be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetraedra. We explain how to produce local coordinates for the variety defined by the gluing equations for SL(3,C)representations. In particular we prove local rigidity of the “geometric” representation in SL(3,C), recovering a recent result of Menal-Ferrer and Porti. More generally we give a criterion for local rigidty of SL(3,C)-representations and provide detailed analysis of the figure eight knot sister manifold exhibiting the different possibilities that can occur.