Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone–3–manifolds

Montcouquiol, Grégoire¹ ; Weiß, Hartmut²

¹ Univ. Paris-Sud, Laboratoire de Mathématiques, UMR8628, CNRS, Orsay F-91405, France
² LMU München, Mathematisches Institut, Theresienstr. 39, D-80333 München, Germany

Geometry & topology, Tome 17 (2013) no. 1, pp. 369-412.

Résumé

In the former articles [arXiv:0903.4743 and this volume pp 329-367], it was independently proven by the authors that the space of hyperbolic cone–$3$–manifolds with cone angles less than $2\pi$ and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, ie when the singular vertices can be split. We show that the different possible splittings correspond to specific pair-of-pants decompositions of the smooth parts of the links of the singular vertices, and that under suitable assumptions the corresponding subspace of deformations is parametrized by the cone angles of the original edges and the lengths of the new ones.

Détail

DOI : 10.2140/gt.2013.17.369

Classification : 57M50, 58D27, 53C35
Keywords: Cone-manifolds, surface group representations, hyperbolic geometry

Bibliographie

[1] M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523

[2] M Boileau, B Leeb, J Porti, Geometrization of 3–dimensional orbifolds, Ann. of Math. 162 (2005) 195

[3] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, 5, Mathematical Society of Japan (2000)

[4] W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200

[5] W M Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986) 263

[6] W M Goldman, Geometric structures on manifolds and varieties of representations, from: "Geometry of group representations" (editors W M Goldman, A R Magid), Contemp. Math. 74, Amer. Math. Soc. (1988) 169

[7] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557

[8] W M Goldman, The complex-symplectic geometry of SL(2, C)–characters over surfaces, from: "Algebraic groups and arithmetic" (editors S G Dani, G Prasad), Tata Inst. Fund. Res. (2004) 375

[9] W J Harvey, Boundary structure of the modular group, from: "Riemann surfaces and related topics : Proceedings of the 1978 Stony Brook Conference" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245

[10] C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1

[11] W P A Klingenberg, Riemannian geometry, 1, Walter de Gruyter Co. (1995)

[12] K Krasnov, J M Schlenker, Minimal surfaces and particles in 3–manifolds, Geom. Dedicata 126 (2007) 187

[13] F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119

[14] H A Masur, Y N Minsky, Geometry of the complex of curves, I, Hyperbolicity, Invent. Math. 138 (1999) 103

[15] R Mazzeo, G Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom. 87 (2011) 525

[16] R Mazzeo, H Weiss, Teichmüller theory for conic surfaces, in preparation

[17] G Montcouquiol, Deformations of hyperbolic convex polyhedra and cone–3–manifolds, to appear in Geom. Dedicata (2013)

[18] G D Mostow, Quasi-conformal mappings in n–space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. (1968) 53

[19] J Porti, Regenerating hyperbolic cone structures from Nil, Geom. Topol. 6 (2002) 815

[20] J Porti, H Weiss, Deforming Euclidean cone 3–manifolds, Geom. Topol. 11 (2007) 1507

[21] J M Schlenker, Dihedral angles of convex polyhedra, Discrete Comput. Geom. 23 (2000) 409

[22] J J Stoker, Geometrical problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1968) 119

[23] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)

[24] W P Thurston, Earthquakes in two-dimensional hyperbolic geometry, from: "Low-dimensional topology and Kleinian groups" (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 91

[25] W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: "The Epstein birthday schrift" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 511

[26] V A Toponogov, Evaluation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR 124 (1959) 282

[27] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793

[28] A Weil, On discrete subgroups of Lie groups, Ann. of Math. 72 (1960) 369

[29] H Weiss, Local rigidity of 3–dimensional cone-manifolds, J. Differential Geom. 71 (2005) 437

[30] H Weiss, Global rigidity of 3–dimensional cone-manifolds, J. Differential Geom. 76 (2007) 495

[31] H Weiss, The deformation theory of hyperbolic cone-3–manifolds with cone-angles less than 2π, Geom. Topol. 17 (2013) 329