Representation varieties detect essential surfaces

Abstract

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 $\operatorname{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $\operatorname{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $\operatorname{SL}_n$-character variety for some $n$.