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

Casella, Alex; Katerba, Charles; Tillmann, Stephan

doi:10.1090/proc/14684

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3–manifolds
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by Alex Casella, Charles Katerba and Stephan Tillmann PDF

Proc. Amer. Math. Soc. 148 (2020), 2257-2271 Request permission

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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