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Regenerating hyperbolic cone 3-manifolds from dimension 2. (Régénérescense des 3-variétés coniques hyperboliques dès la dimension 2.) (English. French summary) Zbl 1293.57012
The space of hyperbolic cone 3-manifolds with fixed topological type and with cone angles less than $$\pi$$ is well understood, but the boundary of this space is not. In the present paper it is shown 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.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)
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##### References:
 [1] Barreto, A. Paiva, Déformation de structures hyperboliques coniques, (2009) [2] Boileau, Michel; Leeb, Bernhard; Porti, Joan, Uniformization of small 3-orbifolds, C. R. Acad. Sci. Paris Sér. I Math., 332, 1, 57-62, (2001) · Zbl 0976.57017 [3] Boileau, Michel; Leeb, Bernhard; Porti, Joan, Geometrization of 3-dimensional orbifolds, Ann. of Math. (2), 162, 1, 195-290, (2005) · Zbl 1087.57009 [4] Boileau, Michel; Porti, Joan, Geometrization of 3-orbifolds of cyclic type, Astérisque, 272, 208 pp., (2001) · Zbl 0971.57004 [5] Bonahon, F.; Siebenmann, L., Low-dimensional topology (Chelwood Gate, 1982), 95, The classification of Seifert fibred $$3$$-orbifolds, 19-85, (1985), Cambridge Univ. Press, Cambridge · Zbl 0571.57011 [6] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P., Three-dimensional orbifolds and cone-manifolds, 5, (2000), Mathematical Society of Japan, Tokyo · Zbl 0955.57014 [7] Culler, Marc, Lifting representations to covering groups, Adv. in Math., 59, 1, 64-70, (1986) · Zbl 0582.57001 [8] Culler, Marc; Shalen, Peter B., Varieties of group representations and splittings of $$3$$-manifolds, Ann. of Math. (2), 117, 1, 109-146, (1983) · Zbl 0529.57005 [9] Danciger, Jeffrey, Geometric transitions: from hyperbolic to AdS geometry, (2011) · Zbl 1287.57020 [10] Fenchel, Werner, Elementary geometry in hyperbolic space, 11, (1989), Walter de Gruyter & Co., Berlin · Zbl 0674.51001 [11] Francaviglia, Stefano, Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not., 9, 425-459, (2004) · Zbl 1088.57015 [12] Goldman, William M., The symplectic nature of fundamental groups of surfaces, Adv. in Math., 54, 2, 200-225, (1984) · Zbl 0574.32032 [13] Goldman, William M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., 85, 2, 263-302, (1986) · Zbl 0619.58021 [14] Goldman, William M., Algebraic groups and arithmetic, The complex-symplectic geometry of $${\rm SL}(2,\mathbb{C})$$-characters over surfaces, 375-407, (2004), Tata Inst. Fund. Res., Mumbai · Zbl 1089.53060 [15] González-Acuña, F.; Montesinos-Amilibia, José María, On the character variety of group representations in $${\rm SL}(2,\textbf{C})$$ and $${\rm PSL}(2,\textbf{C}),$$ Math. Z., 214, 4, 627-652, (1993) · Zbl 0799.20040 [16] Heusener, Michael; Porti, Joan, The variety of characters in $${\rm PSL}_2(\mathbb{C}),$$ Bol. Soc. Mat. Mexicana (3), 10, Special Issue, 221-237, (2004) · Zbl 1100.57014 [17] Hodgson, C., Degeneration and regeneration of hyperbolic structures on three-manifolds, (1986) [18] Kapovich, Michael, Hyperbolic manifolds and discrete groups, 183, (2001), Birkhäuser Boston Inc., Boston, MA · Zbl 1180.57001 [19] Kerckhoff, Steven P., The Nielsen realization problem, Ann. of Math. (2), 117, 2, 235-265, (1983) · Zbl 0528.57008 [20] Lubotzky, Alexander; Magid, Andy R., Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc., 58, 336, xi+117 pp., (1985) · Zbl 0598.14042 [21] Marden, A., Outer circles, (2007), Cambridge University Press, Cambridge · Zbl 1149.57030 [22] Porti, Joan, Regenerating hyperbolic and spherical cone structures from Euclidean ones, Topology, 37, 2, 365-392, (1998) · Zbl 0897.58042 [23] Porti, Joan, Hyperbolic polygons of minimal perimeter with given angles, Geom. Dedicata, 156, 165-170, (2012) · Zbl 1236.51012 [24] Schlenker, Jean-Marc, Small deformations of polygons and polyhedra, Trans. Amer. Math. Soc., 359, 5, 2155-2189, (2007) · Zbl 1126.53041 [25] Weil, André, Remarks on the cohomology of groups, Ann. of Math. (2), 80, 149-157, (1964) · Zbl 0192.12802 [26] Weiss, Hartmut, Local rigidity of 3-dimensional cone-manifolds, J. Differential Geom., 71, 3, 437-506, (2005) · Zbl 1098.53038 [27] Weiss, Hartmut, Global rigidity of 3-dimensional cone-manifolds, J. Differential Geom., 76, 3, 495-523, (2007) · Zbl 1184.53049 [28] Weiss, Hartmut, The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $$2π , (090445682009)$$ · Zbl 1262.53032
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