Undetected boundary slopes and roots of unity for the character variety of a 3-manifold

This dissertation is concerned with the Culler-Shalen techniques for using the SL2(C)-character variety for the fundamental group of a 3-manifold to find embedded essential surfaces in the manifold. It is known that when a boundary slope is strongly detected by the character variety the limiting eigenvalue of the slope is a root of unity. In Chapter 2, we show that every root of unity arises in this manner. Given any root of unity, we construct infinitely many hyperbolic 3-manifolds whose character varieties all detect this root. In Chapter 3, we give infinitely many hyperbolic knots whose exteriors have strict boundary slopes which are not strongly detected by the character variety.