Peripheral birationality for 3-dimensional convex co-compact $PS{L}_2\mathbb{C}$ varieties

Abstract

Let $M$ be a hyperbolizable $3$-manifold with boundary, and let ${\chi }_0(M)$ be a component of the $PS{L}_2\mathbb{C}$-character variety of $M$ that contains the convex co-compact characters. We show that the peripheral map $i_{\ast }:{\chi }_0(M)\to \chi (\partial M)$ to the character variety of $\partial M$ is a birational isomorphism with its image, and in particular is generically a one-to-one map. This generalizes work of Dunfield (one cusped hyperbolic $3$-manifolds) and Klaff-Tillmann (finite volume hyperbolic $3$-manifolds). We use the Bonahon-Schläfli formula and volume rigidity of discrete co-compact representations.

16 pages, 9 figures. Some typos fixed for v.2. Comments are welcome!