Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds
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2020-02
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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.
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Casella, Alex, Charles Katerba, and Stephan Tillmann. “Ideal Points of Character Varieties, Algebraic Non-Integral Representations, and Undetected Closed Essential Surfaces in 3–manifolds.” Proceedings of the American Mathematical Society 148, no. 5 (February 12, 2020): 2257–2271. doi:10.1090/proc/14684.