Mathematical Research Letters

Volume 23 (2016)

Number 4

A birationality result for character varieties

Pages: 1099 – 1110



Ben Klaff (Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Stephan Tillmann (School of Mathematics and Statistics, University of Sydney, NSW, Australia)


Let $M$ be an orientable, cusped hyperbolic $3$-manifold of finite volume. We show that the restriction map $r : \mathfrak{X}_0 \to \mathfrak{X}(\partial M)$ from a Dehn surgery component in the $PSL_2 (\mathbb{C})$-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.

Accepted 13 August 2014

Published 16 September 2016