Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties
Agol, Ian1; Vargas Pallete, Franco2
1Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720 (USA)
2Department of Mathematics, Yale University, 219 Prospect St, Floors 7-9, New Haven, CT 06511 (USA)
Annales de l'Institut Fourier, Online first, 23 p.

Let $M$ be a hyperbolizable $3$-manifold with boundary, and let $\chi _0(M)$ be a component of the $\operatorname{PSL}_2\mathbb{C}$-character variety of $M$ that contains the convex co-compact characters. We show that the peripheral map $i_* \colon \chi _0(M)\rightarrow \chi (\partial M)$ 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–Schläfli formula and volume rigidity of discrete co-compact representations.

Soit $M$ une variété hyperbolisable de dimension $3$ à bord, et soit $\chi _0(M)$ un composant de la variété de caractères $\operatorname{PSL}_2\mathbb{C}$ de $M$ qui contient les caractères co-compacts convexes. Nous montrons que l’application périphérique $i_* \colon \chi _0(M)\rightarrow \chi (\partial M)$ à la variété de caractères de $\partial M$ est un isomorphisme birationnel avec son image, et en particulier est génériquement injective. Cela généralise les travaux de Dunfield (variétés $3$ hyperboliques cuspidées) et de Klaff–Tillmann (variétés $3$ hyperboliques à volumes finis). Nous utilisons la formule de Bonahon–Schläfli et la rigidité volumique des représentations cocompactes discrètes.

Received:
Accepted:
Online First:
DOI: 10.5802/aif.3741
Classification: 57M50, 30F40
Keywords: character variety, hyperbolic volume and rigidity, Culler–Shalen theory
Mots-clés : variété des caractères, volume hyperbolique et rigidité, théorie de Culler–Shalen

Agol, Ian  1 ; Vargas Pallete, Franco  2

1 Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720 (USA)
2 Department of Mathematics, Yale University, 219 Prospect St, Floors 7-9, New Haven, CT 06511 (USA) 
Agol, Ian; Vargas Pallete, Franco. Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties. Annales de l'Institut Fourier, Online first, 23 p.
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     author = {Agol, Ian and Vargas Pallete, Franco},
     title = {Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties},
     journal = {Annales de l'Institut Fourier},
     year = {2025},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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