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

TL;DR

This paper explores the detection of closed essential surfaces in 3-manifolds via character varieties and algebraic representations, providing new examples and resolving open questions in the field.

Contribution

It introduces examples of closed essential surfaces undetectable by ideal points or algebraic non-integral representations, and constructs hyperbolic Haken 3-manifolds lacking such representations.

Findings

Identified closed essential surfaces not detected by existing methods

Constructed hyperbolic Haken 3-manifolds with no algebraic non-integral representations

Resolved a question of Shanuel and Zhang regarding representation detection

Abstract

Closed essential surfaces in a three-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 representations into PSL(2, C), resolving a question of Shanuel and Zhang.