Character varieties, earthquakes and essential surfaces

Abstract

Character varieties of low-dimensional manifolds are a rich area of study that reflect geometric and topological information of the underlying manifold. We study character varieties from two perspectives, with a focus on hyperbolic manifolds. The first perspective uses Teichmüller space and earthquakes. Teichmüller space can be naturally identified with a component of the SL_2(R)–character variety of a surface. We derive explicit forms of the earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus with both algebraic and geometric interpretations. Examining the limiting behaviour gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case. The second perspective uses essential surfaces and the theory of Culler and Shalen. In the seminal work of Culler and Shalen, essential surfaces in 3–manifolds are associated to ideal points of their SL_2(C)–character varieties. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over the SL_2(F)–variety of characters for F an arbitrary algebraically closed field. We provide several applications of the extended theory.