no. 1
Pentagon representations and complex projective structures on closed surfaces
[Représentations pentagones et structures projectives complexes sur les surfaces fermées]
Le Fils, Thomas1
1 Institut de Mathématiques de Jussieu Paris Rive Gauche (IMJ-PRG) 75005 Paris (France)
Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 423-445.

Nous définissons une classe de représentations du groupe fondamental d’une surface fermée de genre 2 dans PSL 2 (ℂ)  : les représentations pentagones. Nous montrons que ce sont exactement les représentations d’un groupe de surface dans PSL 2 (ℂ) qui n’admettent pas de décomposition de Schottky, i.e. de décomposition en pantalons telle que la restriction de la représentation à chaque pantalon est un isomorphisme sur un groupe de Schottky. Ce faisant, nous exhibons une lacune dans la preuve de Gallo, Kapovich et Marden du fait que toute représentation non-élémentaire d’un groupe de surface dans PSL 2 (ℂ) est l’holonomie d’une structure projective, avec éventuellement un point de branchement d’ordre 2. Nous montrons que les représentations pentagones sont de telles holonomies et réparons leur preuve.

We define a class of representations of the fundamental group of a closed surface of genus 2 to PSL 2 (ℂ): the pentagon representations. We show that they are exactly the non-elementary PSL 2 (ℂ)-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to PSL 2 (ℂ) is the holonomy of a projective structure, possibly with one branched point of order 2. We show that pentagon representations arise as such holonomies and repair their proof.

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DOI : 10.5802/aif.3528
Classification : 57M50
Keywords: Projective structure, holonomy, surface groups representations.
Mot clés : Structure projective, holonomie, représentations de groupes de surfaces.
Le Fils, Thomas 1

1 Institut de Mathématiques de Jussieu Paris Rive Gauche (IMJ-PRG) 75005 Paris (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Pentagon representations and complex projective structures on closed surfaces},
     journal = {Annales de l'Institut Fourier},
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Le Fils, Thomas. Pentagon representations and complex projective structures on closed surfaces. Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 423-445. doi : 10.5802/aif.3528. https://aif.centre-mersenne.org/articles/10.5802/aif.3528/

[1] Baba, Shinpei A Schottky decomposition theorem for complex projective structures, Geom. Topol., Volume 14 (2010) no. 1, pp. 117-151 | DOI | MR | Zbl

[2] Benedetto, Robert L.; Goldman, William M. The topology of the relative character varieties of a quadruply-punctured sphere, Experiment. Math., Volume 8 (1999) no. 1, pp. 85-103 http://projecteuclid.org/euclid.em/1047477115 | DOI | MR | Zbl

[3] Deroin, Bertrand; Tholozan, Nicolas Supra-maximal representations from fundamental groups of punctured spheres to PSL(2,ℝ), Ann. Sci. Éc. Norm. Supér. (4), Volume 52 (2019) no. 5, pp. 1305-1329 | DOI | MR | Zbl

[4] Dumas, David Complex projective structures, Handbook of Teichmüller theory. Vol. II (IRMA Lect. Math. Theor. Phys.), Volume 13, Eur. Math. Soc., Zürich, 2009, pp. 455-508 | DOI | MR | Zbl

[5] Earle, Clifford J. On variation of projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Ann. of Math. Stud.), Volume 97, Princeton Univ. Press, Princeton, N.J. (1981), pp. 87-99 | DOI | MR | Zbl

[6] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012, xiv+472 pages | MR | Zbl

[7] Gallo, Daniel; Kapovich, Michael; Marden, Albert The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 625-704 | DOI | MR | Zbl

[8] Goldman, William M. The modular group action on real SL (2)-characters of a one-holed torus, Geom. Topol., Volume 7 (2003), pp. 443-486 | DOI | MR | Zbl

[9] Goldman, William M. What is⋯a projective structure?, Notices Amer. Math. Soc., Volume 54 (2007) no. 1, pp. 30-33 | MR | Zbl

[10] Gupta, Subhojoy Monodromy groups of ℂℙ 1 -structures on punctured surfaces, 2019 (https://arxiv.org/abs/1909.10771)

[11] Hejhal, Dennis A. Monodromy groups and linearly polymorphic functions, Acta Math., Volume 135 (1975) no. 1, pp. 1-55 | DOI | MR | Zbl

[12] Hubbard, John H. The monodromy of projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Ann. of Math. Stud.), Volume 97, Princeton Univ. Press, Princeton, N.J. (1981), pp. 257-275 | DOI | MR | Zbl

[13] Kapovich, Michael Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2009, xxviii+467 pages (reprint of the 2001 edition) | DOI | MR | Zbl

[14] Marché, Julien; Wolff, Maxime Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group, Groups Geom. Dyn., Volume 13 (2019) no. 2, pp. 731-766 | DOI | MR | Zbl

[15] Poincaré, H. Sur les groupes des équations linéaires, Acta Math., Volume 4 (1884) no. 1, pp. 201-312 | DOI | MR | Zbl

[16] Putman, Andrew A note on the connectivity of certain complexes associated to surfaces, Enseign. Math. (2), Volume 54 (2008) no. 3-4, pp. 287-301 | MR | Zbl

[17] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, New York, 2006, xii+779 pages | MR | Zbl

[18] Tan, Ser Peow Branched CP 1 -structures on surfaces with prescribed real holonomy, Math. Ann., Volume 300 (1994) no. 4, pp. 649-667 | DOI | MR | Zbl

[19] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997, x+311 pages (edited by Silvio Levy) | DOI | MR | Zbl

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