Volume of Representations and Birationality of Peripheral Holonomy

ABSTRACT

A theorem by Dunfield states that the peripheral holonomy map from the SL(2, C)-character variety of a 3-manifold to the A-polynomial is a birational isomorphism. We discuss here a possible generalization to SL(n, C)-character varieties. Dunfield's proof involves the rigidity of maximal volume representations and the abundance of hyperbolic Dehn fillings. It is not directly transposable to the SL(n, C)-setting. In this article, we state a conjecture about the volume function on SL(n, C)-character varieties and prove it would imply the generalized birationality result. Some computational experimentations are also described, which support the conjecture.

Keywords:

• character varieties
• volume

2010 AMS Subject Classification:

• 57M27

Notes

1 Recall for example that when n = 3, the representation r_n is also known as the adjoint representation.

2 The reader may as well assume t = 1 and restricts to the case of a knot complement. It will not really interfere, and may simplify notations.

3 The actual function I used on the ideal defined by the gluing equations is “groebner_fan().tropical_intersection().rays()”. When it answers, the answer is a list of points in R^ν, corresponding to the coordinates of the points in the logarithmic limit set of X₂.