Regenerating hyperbolic cone 3-manifolds from dimension 2
Annales de l'Institut Fourier, Volume 63 (2013) no. 5, p. 1971-2015

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regenerations of the polygon, provided that the perimeter is minimal.

On prouve qu’une 3-orbifold close qui fibre sur une 2-orbifold hyperbolique et polygonale admet une famille de structures coniques hyperboliques qu’on voit comme une régénérescence du polygone, pourvu que son périmètre soit minimal.

DOI : https://doi.org/10.5802/aif.2820
Classification:  57M50,  57N10
Keywords: orbifold, hyperbolic cone 3-manifold, degeneration, hyperbolic polygon, perimeter
@article{AIF_2013__63_5_1971_0,
author = {Porti, Joan},
title = {Regenerating hyperbolic cone 3-manifolds from dimension 2},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {63},
number = {5},
year = {2013},
pages = {1971-2015},
doi = {10.5802/aif.2820},
mrnumber = {3186514},
zbl = {1293.57012},
language = {en},
url = {http://www.numdam.org/item/AIF_2013__63_5_1971_0}
}
Porti, Joan. Regenerating hyperbolic cone 3-manifolds from dimension 2. Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 1971-2015. doi : 10.5802/aif.2820. http://www.numdam.org/item/AIF_2013__63_5_1971_0/

[1] Barreto, A. Paiva Déformation de structures hyperboliques coniques (Thèse, Université Paul Sabatier, Toulouse, 2009)

[2] Boileau, Michel; Leeb, Bernhard; Porti, Joan Uniformization of small 3-orbifolds, C. R. Acad. Sci. Paris Sér. I Math., Tome 332 (2001) no. 1, pp. 57-62 | Article | MR 1805628 | Zbl 0976.57017

[3] Boileau, Michel; Leeb, Bernhard; Porti, Joan Geometrization of 3-dimensional orbifolds, Ann. of Math. (2), Tome 162 (2005) no. 1, pp. 195-290 | Article | MR 2178962 | Zbl 1087.57009

[4] Boileau, Michel; Porti, Joan Geometrization of 3-orbifolds of cyclic type, Astérisque (2001) no. 272, pp. 208 (Appendix A by Michael Heusener and Porti) | MR 1844891

[5] Bonahon, F.; Siebenmann, L. The classification of Seifert fibred $3$-orbifolds, Low-dimensional topology (Chelwood Gate, 1982), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 95 (1985), pp. 19-85 | MR 827297 | Zbl 0571.57011

[6] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three-dimensional orbifolds and cone-manifolds, Mathematical Society of Japan, Tokyo, MSJ Memoirs, Tome 5 (2000) (With a postface by Sadayoshi Kojima) | MR 1778789 | Zbl 0955.57014

[7] Culler, Marc Lifting representations to covering groups, Adv. in Math., Tome 59 (1986) no. 1, pp. 64-70 | Article | MR 825087 | Zbl 0582.57001

[8] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. (2), Tome 117 (1983) no. 1, pp. 109-146 | Article | MR 683804 | Zbl 0529.57005

[9] Danciger, Jeffrey Geometric transitions: from hyperbolic to AdS geometry (Thesis, Stanford University, 2011)

[10] Fenchel, Werner Elementary geometry in hyperbolic space, Walter de Gruyter & Co., Berlin, de Gruyter Studies in Mathematics, Tome 11 (1989) (With an editorial by Heinz Bauer) | MR 1004006 | Zbl 0674.51001

[11] Francaviglia, Stefano Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not. (2004) no. 9, pp. 425-459 | Article | MR 2040346 | Zbl 1088.57015

[12] Goldman, William M. The symplectic nature of fundamental groups of surfaces, Adv. in Math., Tome 54 (1984) no. 2, pp. 200-225 | Article | MR 762512 | Zbl 0574.32032

[13] Goldman, William M. Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., Tome 85 (1986) no. 2, pp. 263-302 | Article | MR 846929 | Zbl 0619.58021

[14] Goldman, William M. The complex-symplectic geometry of $\mathrm{SL}\left(2,ℂ\right)$-characters over surfaces, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai (2004), pp. 375-407 | MR 2094117 | Zbl 1089.53060

[15] González-Acuña, F.; Montesinos-Amilibia, José María On the character variety of group representations in $\mathrm{SL}\left(2,\mathbf{C}\right)$ and $\mathrm{PSL}\left(2,\mathbf{C}\right)$, Math. Z., Tome 214 (1993) no. 4, pp. 627-652 | Article | MR 1248117 | Zbl 0799.20040

[16] Heusener, Michael; Porti, Joan The variety of characters in ${\mathrm{PSL}}_{2}\left(ℂ\right)$, Bol. Soc. Mat. Mexicana (3), Tome 10 (2004) no. Special Issue, pp. 221-237 | MR 2199350 | Zbl 1100.57014

[17] Hodgson, C. Degeneration and Regeneration of Hyperbolic Structures on Three-Manifolds (Thesis, Princeton University, 1986)

[18] Kapovich, Michael Hyperbolic manifolds and discrete groups, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 183 (2001) | MR 1792613 | Zbl 1180.57001

[19] Kerckhoff, Steven P. The Nielsen realization problem, Ann. of Math. (2), Tome 117 (1983) no. 2, pp. 235-265 | Article | MR 690845 | Zbl 0528.57008

[20] Lubotzky, Alexander; Magid, Andy R. Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc., Tome 58 (1985) no. 336, pp. xi+117 | MR 818915 | Zbl 0598.14042

[21] Marden, A. Outer circles, Cambridge University Press, Cambridge (2007) (An introduction to hyperbolic 3-manifolds) | Article | MR 2355387 | Zbl 1149.57030

[22] Porti, Joan Regenerating hyperbolic and spherical cone structures from Euclidean ones, Topology, Tome 37 (1998) no. 2, pp. 365-392 | Article | MR 1489209 | Zbl 0897.58042

[23] Porti, Joan Hyperbolic polygons of minimal perimeter with given angles, Geom. Dedicata, Tome 156 (2012), pp. 165-170 | Article | MR 2863552 | Zbl 1236.51012

[24] Schlenker, Jean-Marc Small deformations of polygons and polyhedra, Trans. Amer. Math. Soc., Tome 359 (2007) no. 5, pp. 2155-2189 | Article | MR 2276616 | Zbl 1126.53041

[25] Weil, André Remarks on the cohomology of groups, Ann. of Math. (2), Tome 80 (1964), pp. 149-157 | Article | MR 169956 | Zbl 0192.12802

[26] Weiss, Hartmut Local rigidity of 3-dimensional cone-manifolds, J. Differential Geom., Tome 71 (2005) no. 3, pp. 437-506 | MR 2198808 | Zbl 1098.53038

[27] Weiss, Hartmut Global rigidity of 3-dimensional cone-manifolds, J. Differential Geom., Tome 76 (2007) no. 3, pp. 495-523 http://projecteuclid.org/getRecord?id=euclid.jdg/1180135696 | MR 2331529 | Zbl 1184.53049

[28] Weiss, Hartmut The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$ (arXiv:0904.4568, 2009, to appear in Geom. and Top) | MR 3035330 | Zbl 1262.53032