Normal surfaces, angle structures and the character variety

I describe geometric splittings of cusped hyperbolic 3-manifolds associated to ideal points of the character variety using elementary and (almost) computational methods. Character varieties allow the study of representations into SL(2,C) and PSL(2,C) up to conjugacy, and provide a link between the topology and the geometry of a 3-manifold. I describe this link for cusped hyperbolic 3-manifolds with suitable ideal triangulations. The techniques used are geometric and combinatorial; normal surface theory, standard spines, angle structures and hyperbolic geometry are involved.