×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI arXiv
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.