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arXiv · 2411.06859

An invitation to Culler-Shalen theory in arbitrary characteristic

Grace S. Garden·Stephan Tillmann
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Abstract

In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-character varieties, and connections between the algebraic geometry of the character variety and the topology of the 3-manifold are established via group actions on trees. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over an arbitrary algebraically closed field. Examples include a change in the $A$-polynomial in characteristic 2, a closed Haken hyperbolic 3-manifold with no detected essential surface, and a closed Haken hyperbolic 3-manifold with an essential surface only detected in characteristic 2.

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Grace S. Garden, Stephan Tillmann. 2024-12-12. An invitation to Culler-Shalen theory in arbitrary characteristic. https://arxiv.org/abs/2411.06859

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