An invitation to Culler-Shalen theory in arbitrary characteristic

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.