Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds

Abstract

Closed essential surfaces in a 3-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representation into $ \textrm {PSL}_2 (\mathbb{C})$, resolving a question of Schanuel and Zhang.