Peripheral birationality for 3-dimensional convex co-compact $PSL_2\mathbb{C}$ varieties

TL;DR

This paper proves that for certain hyperbolic 3-manifolds, the peripheral character map is a birational isomorphism, extending previous results and using volume rigidity and the Bonahon-Schläfli formula.

Contribution

It establishes the peripheral map as a birational isomorphism for hyperbolizable 3-manifolds, generalizing prior work on specific hyperbolic 3-manifolds.

Findings

Peripheral map is a birational isomorphism with its image.

The result generalizes previous work by Dunfield and Klaff-Tillmann.

Uses Bonahon-Schläfli formula and volume rigidity in the proof.

Abstract

Let $M$ be a hyperbolizable $3$-manifold with boundary, and let $\chi_0(M)$ be a component of the $PSL_2\mathbb{C}$-character variety of $M$ that contains the convex co-compact characters. We show that the peripheral map $i_*:\chi_0(M)\rightarrow\chi(\partial M)$ 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-Schl\"afli formula and volume rigidity of discrete co-compact representations.