Supra-maximal representations from fundamental groups of punctured spheres to  PSL (2,ℝ)
[Représentations supra-maximales des groupes fondamentaux de sphères épointées dans PSL (2,ℝ).]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1305-1329.
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Nous étudions une classe particulière de représentations du groupe fondamental des sphères épointées Σ0,n dans le groupe PSL (2,ℝ), que nous appelons supra-maximales. Bien qu'elles soient pour la plupart Zariski denses, nous montrons qu'elles sont totalement non hyperboliques, au sens où l'image de toute courbe fermée simple est elliptique ou parabolique. Nous montrons aussi qu'elles sont géométrisables (hormis celles qui sont réductibles) en un sens très fort : pour tout élément de l'espace de Teichmüller 𝒯0,n, il existe une unique application équivariante holomorphe à valeurs dans le demi-plan inférieur ℍ-. Nous montrons également que les représentations supra-maximales forment des composantes compactes des variétés de caractère relatives. Munies de la structure symplectique de Atiyah-Bott-Goldman, ces composantes sont symplectomorphes à l'espace projectif complexe de dimension n-3 muni d'un multiple de la forme de Fubini-Study que nous calculons explicitement. Cela généralise un résultat de Benedetto-Goldman pour la sphère à quatre trous.

We study a particular class of representations from the fundamental groups of punctured spheres Σ0,n to the group PSL(2,ℝ), which we call supra-maximal. Though most of them are Zariski dense, we show that supra-maximal representations are totally non hyperbolic, in the sense that every simple closed curve is mapped to an elliptic or parabolic element. They are also shown to be geometrizable (apart from the reducible ones) in the following very strong sense : for any element of the Teichmüller space 𝒯0,n, there is a unique holomorphic equivariant map with values in the lower half-plane ℍ-. In the relative character varieties, the components of supra-maximal representations are shown to be compact and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension n-3 equipped with a certain multiple of the Fubini-Study form that we compute explicitly. This generalizes a result of Benedetto-Goldman [3] for the sphere minus four points.

Publié le :
DOI : 10.24033/asens.2410 
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     author = {Deroin, Bertrand and Tholozan, Nicolas},
     title = {Supra-maximal representations  from fundamental groups  of punctured spheres to~$\mathrm {PSL} (2,\mathbb {R})$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1305--1329},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {5},
     year = {2019},
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Deroin, Bertrand; Tholozan, Nicolas. Supra-maximal representations  from fundamental groups  of punctured spheres to $\mathrm {PSL} (2,\mathbb {R})$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1305-1329. doi : 10.24033/asens.2410. http://www.numdam.org/articles/10.24033/asens.2410/

Atiyah, M. F.; Bott, R. The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, Volume 308 (1983), pp. 523-615 (ISSN: 0080-4614) | DOI | MR | Zbl

Basmajian, A. Universal length bounds for non-simple closed geodesics on hyperbolic surfaces, J. Topol., Volume 6 (2013), pp. 513-524 (ISSN: 1753-8416) | DOI | MR | Zbl

Benedetto, R. L.; Goldman, W. M. The topology of the relative character varieties of a quadruply-punctured sphere, Experiment. Math., Volume 8 (1999), pp. 85-103 http://projecteuclid.org/euclid.em/1047477115 (ISSN: 1058-6458) | DOI | MR | Zbl

Burger, M.; Iozzi, A. Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity, Geom. Dedicata, Volume 125 (2007), pp. 1-23 (ISSN: 0046-5755) | DOI | MR | Zbl

Burger, M.; Iozzi, A.; Wienhard, A. Surface group representations with maximal Toledo invariant, Ann. of Math., Volume 172 (2010), pp. 517-566 (ISSN: 0003-486X) | DOI | MR | Zbl

Bowditch, B. H. Markov triples and quasi-Fuchsian groups, Proc. London Math. Soc., Volume 3 (1998), pp. 697-736 | DOI | MR | Zbl

Delzant, T. Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, Volume 116 (1988), pp. 315-339 | DOI | Numdam | MR | Zbl

Ghys, É. Groups acting on the circle, Enseign. Math., Volume 47 (2001), pp. 329-407 (ISSN: 0013-8584) | MR | Zbl

Guéritaud, F.; Kassel, F. Maximally stretched laminations on geometrically finite hyperbolic manifolds, Geom. Topol., Volume 21 (2017), pp. 693-840 (ISSN: 1465-3060) | DOI | MR | Zbl

Gallo, D.; Kapovich, M.; Marden, A. The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math., Volume 151 (2000), pp. 625-704 (ISSN: 0003-486X) | DOI | MR | Zbl

Goldman, W. M. The modular group action on real SL (2)–characters of a one-holed torus, Geometry & Topology, Volume 7 (2003), pp. 443-486 | DOI | MR | Zbl

Goldman, W. M. Mapping class group dynamics on surface group representations. Problems on mapping class groups and related topics, 189–214, Proc. Sympos. Pure Math, Volume 74 (2006) | DOI | MR | Zbl

Goldman, W. M. Discontinuous groups and the Euler class, ProQuest LLC, Ann Arbor, MI (1980) http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:8029408 | MR

Goldman, W. M. The symplectic nature of fundamental groups of surfaces, Adv. in Math., Volume 54 (1984), pp. 200-225 (ISSN: 0001-8708) | DOI | MR | Zbl

Goldman, W. M. Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. math., Volume 85 (1986), pp. 263-302 (ISSN: 0020-9910) | DOI | MR | Zbl

Goldman, W. M. Topological components of spaces of representations, Invent. math., Volume 93 (1988), pp. 557-607 (ISSN: 0020-9910) | DOI | MR | Zbl

Goldman, W. M. Ergodic theory on moduli spaces, Ann. of Math., Volume 146 (1997), pp. 475-507 (ISSN: 0003-486X) | DOI | MR | Zbl

Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc., Volume 55 (1987), pp. 59-126 (ISSN: 0024-6115) | DOI | MR | Zbl

Koziarz, V.; Maubon, J. Harmonic maps and representations of non-uniform lattices of PU (m,1) , Ann. Inst. Fourier, Volume 58 (2008), pp. 507-558 http://aif.cedram.org/item?id=AIF_2008__58_2_507_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

McOwen, R. C. Point Singularities and Conformal Metrics on Riemann Surfaces, Proceedings of the American Mathematical Society, Volume 103 (1988), pp. 222-224 http://www.jstor.org/stable/2047555 | DOI | MR | Zbl

Milnor, J. On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223 (ISSN: 0010-2571) | DOI | MR | Zbl

Mondello, G., In the tradition of Ahlfors-Bers. V (Contemp. Math.), Volume 510, Amer. Math. Soc., 2010, pp. 307-329 | DOI | MR | Zbl

Mondello, G. Topology of representation spaces of surface groups in PSL (2,ℝ) with assigned boundary monodromy, Pure and Applied Mathematics Quarterly, Volume 12 (2016), pp. 399-462 | DOI | MR | Zbl

Marché, J.; Wolff, M. The modular action on PSL 2(ℝ)-characters in genus 2, Duke Math. J., Volume 165 (2016), pp. 371-412 | DOI | MR | Zbl

Tan, S. P. Branched 𝐂P1-structures on surfaces with prescribed real holonomy, Math. Ann., Volume 300 (1994), pp. 649-667 (ISSN: 0025-5831) | DOI | MR | Zbl

Troyanov, M. Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., Volume 324 (1991), pp. 793-821 (ISSN: 0002-9947) | DOI | MR | Zbl

Wood, J. W. Bundles with totally disconnected structure group, Comment. Math. Helv., Volume 46 (1971), pp. 257-273 (ISSN: 0010-2571) | DOI | MR | Zbl

Yang, T. On type-preserving representations of the four-punctured sphere group, Geom. Topol., Volume 20 (2016), pp. 1213-1255 | DOI | MR | Zbl

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