Representation varieties detect essential surfaces

Stefan Friedl; Takahiro Kitayama; Matthias Nagel

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arXiv · 1604.00584

Representation varieties detect essential surfaces

Stefan Friedl·Takahiro Kitayama·Matthias Nagel

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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 $n$.

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Stefan Friedl, Takahiro Kitayama, Matthias Nagel. 2016-04-03. Representation varieties detect essential surfaces. https://arxiv.org/abs/1604.00584

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