Steven Boyer - Norm duality and hyperbolic 3-manifolds

The SL₂(C)-character variety of a 1-cusped hyperbolic 3-manifold has given rise to two convex, balanced plane polygons--the unit ball of the Culler-Shalen norm and the Newton polygon of the (two-variable) A-polynomial of the manifold. In joint work with Xingru Zhang, we show that these polygons are dual in the sense that the line segments joining antipodal vertices of one are parallel to the the sides of the other. Indeed, we show that the Newton polygon can be thought of as a ball of the norm dual to the Culler-Shalen norm. This duality was one of the key ideas in our recent proof of the finite surgery conjecture.