Peripheral birationality for 3-dimensional convex co-compact PSL 2 ℂ varieties
[Birationalité périphérique pour les PSL 2 ℂ variétés convexes-cocompactes tridimensionnelles]
Agol, Ian1 ; Vargas Pallete, Franco2
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)
Annales de l'Institut Fourier, Online first, 23 p.

Let M be a hyperbolizable 3-manifold with boundary, and let χ 0 (M) be a component of the PSL 2 ℂ-character variety of M that contains the convex co-compact characters. We show that the peripheral map i * :χ 0 (M)→χ(∂M) to the character variety of ∂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 χ 0 (M) un composant de la variété de caractères PSL 2 ℂ de M qui contient les caractères co-compacts convexes. Nous montrons que l’application périphérique i * :χ 0 (M)→χ(∂M) à la variété de caractères de ∂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.

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Accepté le :
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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) 
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     title = {Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties},
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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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