A construction of representations of $3$-manifold groups into $\mathrm{PU}(2,1)$ through Lefschetz fibrations

Dashyan, Ruben

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 4, pp. 769-803

We obtain infinitely many (non-conjugate) representations of $3$-manifold fundamental groups into a lattice in $\mathrm{Isom}\left({ℍ}_{ℂ}^2\right)$, the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The $3$-manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.

Nous obtenons un nombre infini de représentations non-conjuguées de groupes fondamentaux de $3$-variétés dans un réseau du groupe $\mathrm{Isom}\left({ℍ}_{ℂ}^2\right)$ des isométries holomorphes de l’espace hyperbolique complexe. Ce réseau est un groupe fondamental orbifold d’un revêtement du plan projectif ramifié le long d’un arrangement de droites, construit par Hirzebruch. Les $3$-variétés sont liées à une fibration de Lefschetz de l’orbifold hyperbolique complexe.

[1] Burns, Daniel Jr; Shnider, Steven Spherical hypersurfaces in complex manifolds, Invent. Math., Volume 33 (1976), pp. 223-246 | Zbl | DOI | MR

[2] Cartan, Élie Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, Ann. Mat. Pura Appl., Volume 11 (1932), pp. 17-90 | DOI

[3] Casson, Andrew J.; Bleiler, Steven A. Automorphisms of surfaces after Nielsen and Thurston., London Mathematical Society Student Texts, 9, Cambridge University Press, 1988, 105 pages | Zbl | DOI

[4] Dashyan, Ruben Representations of fundamental groups in hyperbolic geometry, Ph. D. Thesis, Université Pierre et Marie Curie - Paris VI (2017) (https://theses.hal.science/tel-01684245)

[5] Deraux, Martin Forgetful maps between Deligne–Mostow ball quotients, Geom. Dedicata, Volume 150 (2011), pp. 377-389 | Zbl | DOI | MR

[6] Deraux, Martin; Falbel, Elisha Complex hyperbolic geometry of the figure-eight knot, Geom. Topol., Volume 19 (2015) no. 1, pp. 237-293 | DOI | Zbl | MR

[7] Dolgachev, Igor Lectures on invariant theory, Cambridge University Press, 2003, xvi+220 pages | Zbl | DOI

[8] Falbel, Elisha A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differ. Geom., Volume 79 (2008) no. 1, pp. 69-110 | DOI | Zbl | MR

[9] Falbel, Elisha; Wang, Jieyan Branched spherical CR structures on the complement of the figure-eight knot, Mich. Math. J., Volume 63 (2014) no. 3, pp. 635-667 | DOI | Zbl | MR

[10] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton University Press, 2011, xiv+492 pages | Zbl | DOI

[11] Travaux de Thurston sur les surfaces. Séminaire Orsay (Fathi, Albert; Laudenbach, François; Poénaru, Valentin, eds.), Astérisque, 66-67, Société Mathématique de France, 1991, 286 pages | Zbl

[12] Hirzebruch, Friedrich Arrangements of lines and algebraic surfaces, Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. II: Geometry (Progress in Mathematics), Volume 36, Birkhäuser, 1983, pp. 113-140 | Zbl | MR

[13] McMullen, Curtis T. Braid groups and Hodge theory, Math. Ann., Volume 355 (2013) no. 3, pp. 893-946 | Zbl | MR | DOI

[14] Miyaoka, Yoichi Algebraic surfaces with positive indices, Classification of algebraic and analytic manifolds (Proc. Symp. Katata/Jap. 1982) (Progress in Mathematics), Volume 39, Birkhäuser, 1983, pp. 281-301 | Zbl | MR

[15] Namba, Makoto Branched coverings and algebraic functions, Pitman Research Notes in Mathematics Series, 161, Longman Scientific & Technical; John Wiley & Sons, 1987, 201 pages | Zbl

[16] Otal, Jean-Pierre Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque, 235, Société Mathématique de France, 1996, x+159 pages | Zbl | MR | Numdam

[17] Parker, John R. Complex hyperbolic lattices, Discrete groups and geometric structures. Proceedings of the 5th workshop on discrete groups and geometric structures, with applications III, Kortrijk, Belgium, May 26–30, 2008 (Contemporary Mathematics), Volume 501, American Mathematical Society, 2009, pp. 1-42 | Zbl | MR

[18] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc., Volume 19 (1988) no. 2, pp. 417-431 | Zbl | DOI | MR

[19] Tretkoff, Paula Complex ball quotients and line arrangements in the projective plane, Mathematical Notes (Princeton), 51, Princeton University Press, 2016, ix+215 pages (with an appendix by Hans-Christoph Im Hof) | DOI | Zbl | MR

[20] Yamazaki, Tadashi; Yoshida, Masaaki On Hirzebruch’s examples of surfaces with $c_1^2=3c_2$, Math. Ann., Volume 266 (1984), pp. 421-431 | Zbl | DOI | MR