Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

Acosta, Miguel¹

¹ Unité de recherche en Mathématiques, Université du Luxembourg, Campus Belval, Esch-sur-Alzette, Luxembourg

Geometry & topology, Tome 23 (2019) no. 5, pp. 2593-2664.

Résumé

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as a starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane ${ℍ}_{ℂ}^2$. We deform the Ford domain of Parker and Will in ${ℍ}_{ℂ}^2$ in a one-parameter family. On one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer $n\ge 4$, and the spherical CR structure obtained for $n=4$ is the Deraux–Falbel spherical CR uniformization of the figure eight knot complement.

Détail

DOI : 10.2140/gt.2019.23.2593

Classification : 51M10, 57M50, 22E40, 32V05
Keywords: spherical CR geometry, complex hyperbolic geometry, Dehn surgery, uniformization, Whitehead link, Ford domain

Bibliographie

[1] M Acosta, Spherical CR Dehn surgeries, Pacific J. Math. 284 (2016) 257 | DOI

[2] M Acosta, Character varieties for real forms, Geom. Dedicata (2019) | DOI

[3] M Deraux, Deforming the R–Fuchsian (4,4,4)–triangle group into a lattice, Topology 45 (2006) 989 | DOI

[4] M Deraux, On spherical CR uniformization of 3–manifolds, Exp. Math. 24 (2015) 355 | DOI

[5] M Deraux, A 1–parameter family of spherical CR uniformizations of the figure eight knot complement, Geom. Topol. 20 (2016) 3571 | DOI

[6] M Deraux, E Falbel, Complex hyperbolic geometry of the figure-eight knot, Geom. Topol. 19 (2015) 237 | DOI

[7] E Falbel, A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differential Geom. 79 (2008) 69 | DOI

[8] E Falbel, A Guilloux, P V Koseleff, F Rouillier, M Thistlethwaite, Character varieties for SL(3, C) : the figure eight knot, Exp. Math. 25 (2016) 219 | DOI

[9] E Falbel, N Gusevskii, Spherical CR-manifolds of dimension 3, Bol. Soc. Brasil. Mat. 25 (1994) 31 | DOI

[10] E Falbel, P V Koseleff, F Rouillier, Representations of fundamental groups of 3–manifolds into PGL(3, C) : exact computations in low complexity, Geom. Dedicata 177 (2015) 229 | DOI

[11] W M Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3–manifolds, Trans. Amer. Math. Soc. 278 (1983) 573 | DOI

[12] W M Goldman, Complex hyperbolic geometry, Clarendon (1999)

[13] W M Goldman, J R Parker, Complex hyperbolic ideal triangle groups, J. Reine Angew. Math. 425 (1992) 71 | DOI

[14] A Guilloux, P Will, On SL(3, C)–representations of the Whitehead link group, Geom. Dedicata 202 (2019) 81 | DOI

[15] A Lubotzky, A R Magid, Varieties of representations of finitely generated groups, 336, Amer. Math. Soc. (1985) | DOI

[16] J R Parker, Complex hyperbolic Kleinian groups, (2020)

[17] J R Parker, J Wang, B Xie, Complex hyperbolic (3,3,n) triangle groups, Pacific J. Math. 280 (2016) 433 | DOI

[18] J R Parker, P Will, A complex hyperbolic Riley slice, Geom. Topol. 21 (2017) 3391 | DOI

[19] R E Schwartz, Ideal triangle groups, dented tori, and numerical analysis, Ann. of Math. 153 (2001) 533 | DOI

[20] R E Schwartz, Complex hyperbolic triangle groups, from: "Proceedings of the International Congress of Mathematicians, II" (editor T Li), Higher Ed. (2002) 339

[21] R E Schwartz, A better proof of the Goldman–Parker conjecture, Geom. Topol. 9 (2005) 1539 | DOI

[22] R E Schwartz, Spherical CR geometry and Dehn surgery, 165, Princeton Univ. Press (2007) | DOI

[23] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[24] P Will, Two-generator groups acting on the complex hyperbolic plane, from: "Handbook of Teichmüller theory, VI" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 27, Eur. Math. Soc. (2016) 275