Representation varieties detect essential surfaces

Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a 3-manifold from an ideal point of a curve in the SLn-character variety. There exists an essential surface in some 3-manifold known to be not detected in the classical SL2-theory. We prove that every connected essential surface in a 3-manifold is given by an ideal point of a rational curve in the SLn-character variety for some n