Volumes of representations and birationality of the peripheral holonomy

TL;DR

This paper explores the relationship between volume functions and the birationality of the peripheral holonomy map in 3-manifold character varieties, proposing a conjecture that could extend Dunfield's theorem.

Contribution

It introduces a conjecture linking volume functions to the birationality of the peripheral holonomy map, generalizing Dunfield's theorem without a complete proof.

Findings

Proposes a conjecture relating volume functions to birationality.

Highlights the importance of volume in understanding character varieties.

Extends Dunfield's theorem under a new conjectural framework.

Abstract

We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The volume is still an important ingredient in this paper. Unfortunately at this point no complete proof is done. Instead, a conjecture is stated about the volume function on the character variety that would imply the generalized birationality result.