In the seminal work of Culler and Shalen (1983) a method is outlined to detect essential surfaces in a three-manifold by studying their SL(2,C)-character variety. The method underscores connections between the theory of incompressible surfaces in three-manifolds, splittings of fundamental groups, group actions on trees, and the geometry of representation varieties. In this talk, we will motivate and then lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over F, an algebraically closed field of positive characteristic. We then apply the extended theory to a variety of settings. This is joint work with Stephan Tillmann.
