A birationality result for character varieties

TL;DR

This paper proves that for certain hyperbolic 3-manifolds, the restriction map from a Dehn surgery component of the character variety to the boundary's character variety is a birational isomorphism, generalizing Dunfield's result.

Contribution

It establishes a birationality result for the restriction map in character varieties of hyperbolic 3-manifolds, extending previous work by Dunfield.

Findings

The restriction map is a birational isomorphism onto its image.

The volume differential on the eigenvalue variety is exact.

Generalizes Dunfield's earlier results.

Abstract

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.