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) 
@unpublished{AIF_0__0_0_A29_0,
     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},
     doi = {10.5802/aif.3741},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Agol, Ian
AU  - Vargas Pallete, Franco
TI  - Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3741
LA  - en
ID  - AIF_0__0_0_A29_0
ER  - 
%0 Unpublished Work
%A Agol, Ian
%A Vargas Pallete, Franco
%T Peripheral birationality for 3-dimensional convex co-compact $\operatorname{PSL}_2\mathbb{C}$ varieties
%J Annales de l'Institut Fourier
%D 2025
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3741
%G en
%F AIF_0__0_0_A29_0
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.

[1] Ahlfors, Lars; Bers, Lipman Riemann’s mapping theorem for variable metrics, Ann. Math. (2), Volume 72 (1960), pp. 385-404 | DOI | MR | Zbl

[2] Bonahon, Francis Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form, Ann. Fac. Sci. Toulouse, Math. (6), Volume 5 (1996) no. 2, pp. 233-297 | MR | DOI | Numdam | Zbl

[3] Bonahon, Francis Transverse Hölder distributions for geodesic laminations, Topology, Volume 36 (1997) no. 1, pp. 103-122 | DOI | MR | Zbl

[4] Bonahon, Francis A Schläfli-type formula for convex cores of hyperbolic 3-manifolds, J. Differ. Geom., Volume 50 (1998) no. 1, pp. 25-58 | MR | Zbl

[5] Boyer, S.; Zhang, X. On Culler–Shalen seminorms and Dehn filling, Ann. Math. (2), Volume 148 (1998) no. 3, pp. 737-801 | DOI | MR | Zbl

[6] Brooks, Robert Circle packings and co-compact extensions of Kleinian groups, Invent. Math., Volume 86 (1986) no. 3, pp. 461-469 | DOI | MR | Zbl

[7] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. Math. (2), Volume 117 (1983) no. 1, pp. 109-146 | DOI | MR | Zbl

[8] Dumas, David; Kent, Autumn Slicing, skinning, and grafting, Am. J. Math., Volume 131 (2009) no. 5, pp. 1419-1429 | DOI | MR | Zbl

[9] Dunfield, Nathan M. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math., Volume 136 (1999) no. 3, pp. 623-657 | DOI | MR | Zbl

[10] Francaviglia, Stefano Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not. (2004) no. 9, pp. 425-459 | DOI | MR | Zbl

[11] Francaviglia, Stefano; Klaff, Ben Maximal volume representations are Fuchsian, Geom. Dedicata, Volume 117 (2006), pp. 111-124 | DOI | MR | Zbl

[12] Harris, Joe Algebraic geometry, Graduate Texts in Mathematics, 133, Springer, 1992, xx+328 pages | DOI | MR | Zbl

[13] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | MR | DOI | Zbl

[14] Klaff, Ben; Tillmann, Stephan A birationality result for character varieties, Math. Res. Lett., Volume 23 (2016) no. 4, pp. 1099-1110 | DOI | MR | Zbl

[15] Thurston, William P. The geometry and topology of three-manifolds (2002) (online at https://library.slmath.org/nonmsri/gt3m/)

Cited by Sources: