Astérisque MICHEL BOILEAU JOAN PORTI Geometrization of 3-orbifolds of cyclic type Astérisque, tome 272 (2001), p. I-208 <http://www.numdam.org/item?id=AST_2001__272__R1_0> © Société mathématique de France, 2001, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ASTÉRISQUE 272 GEOMETRIZATION OF 3-ORBIFOLDS OF CYCLIC TYPE Michel Boileau and Joan Porti with an Appendix : Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti Société Mathématique de France 2001 Publié avec le concours du Centre National de la Recherche Scientifique M. Boileau Laboratoire Emile Picard, CNRS UMR 5580, Université Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 4, France. E-mail : [email protected] J. Porti Départament de Matemàtiques, Universität Autônoma de Barcelona, E-08193 Bellaterra, Spain. E-mail : porti@mat. uab. es Michael Heusener Laboratoire de Mathématiques Pures, Université Biaise Pascal, F-63177 Aubière Cedex, France. E-mail : heusener@ucfma. univ-bpclermont. f r 2000 Mathematics Subject Classification. — 57M50, 57M60, 53C20, 53C23. Key words and phrases. — Orbifold, hyperbolic, cone manifold, collapse, simplicial volume, Kleinian group. Both authors have been strongly supported by the European TMR ERBFMRXCT 960040 Singularidades de ecuaciones diferenciales y foliaciones. The second named author was partially supported by DGICT through project PB96-1152. GEOMETRIZATION OF 3-ORBIFOLDS OF CYCLIC TYPE Michel Boileau and Joan Porti with the collaboration of Michael Heusener Abstract. — We prove the orbifold theorem in the cyclic case: If O is a compact oriented irreducible atoroidal 3-orbifold whose ramification locus is a non-empty sub- manifold, then O is geometric, i.e. it has a hyperbolic, a Euclidean or a Seifert fibred structure. This theorem implies Thurston's geometrization conjecture for compact orientable irreducible three-manifolds having a non-free symmetry. Résumé (Géométrisation des orbi-variétés tridimensionnelles de type cyclique) Nous démontrons le théorème des orbi-variétés de Thurston dans le cas cyclique : une orbi-variété tridimensionelle, compacte, orientable, irréductible, atoroïdale et dont le lieu de ramification est une sous-variété non vide, admet soit une structure hyper­ bolique ou Euclidienne, soit une fibration de Seifert. Ce théorème implique qu'une variété tridimensionelle, compacte, irréductible et possédant une symétrie non libre, vérifie la conjecture de géométrisation de Thurston. © Astérisque 272, SMF 2001 CONTENTS Introduction 1 1. Cone Manifolds 9 1.1. Basic Definitions 9 1.2. Sequences of hyperbolic cone 3-manifolds 11 2. Proof of Thurston's orbifold theorem for very good 3-orbifolds .... 15 2.1. Generalized Hyperbolic Dehn Filling 15 2.2. The space of hyperbolic cone structures 16 2.3. Proof of Theorem 4 from Theorems A and B 19 2.4. Proof of Theorem 1 23 3. A compactness theorem for cone 3-manifolds with cone angles bounded above by 7r 33 3.1. The Dirichlet polyhedron 34 3.2. Hausdorff-Gromov convergence for cone 3-manifolds 39 3.3. Hausdorff-Gromov convergence implies geometric convergence 41 3.4. Uniform lower bound for the cone-injectivity radius 47 3.5. Some properties of geometric convergence 54 3.6. Cone 3-manifolds with totally geodesic boundary 56 4. Local soul theorem for cone 3-manifolds with cone angles less than or equal to IT 61 4.1. Thurston's classification theorem of non-compact Euclidean cone 3-manifolds 63 4.2. Totally s-convex subsets in Euclidean cone 3-manifolds 65 4.3. Proof of the soul theorem 72 4.4. Proof of the local soul theorem 75 4.5. Local soul theorem for cone 3-manifolds with boundary 77 5. Sequences of closed hyperbolic cone 3-manifolds with cone angles less than IT 79 5.1. The non-collapsing case 80 5.2. The collapsing case 84 vi CONTENTS 5.3. Coverings a la Gromov 86 5.4. From (e, D)-Margulis' coverings of abelian type to 77-coverings a la Gromov 93 5.5. (e, D)-Margulis' neighborhood of thick turnover type 98 6. Very good orbifolds and sequences of hyperbolic cone 3-manifolds .. 103 6.1. The non-collapsing case 103 6.2. The collapsing case 106 6.3. From (e, jD)-Margulis' coverings of type a) and b) to 77-coverings a la Gromov 112 7. Uniformization of small 3-orbifolds 119 7.1. Desingularization of ramified circle components 120 7.2. Uniformization of small 3-orbifolds with non-empty boundary 122 7.3. The sequence does not collapse 124 8. Haken 3-orbifolds 131 8.1. Proof of Thurston's Orbifold theorem 132 8.2. Fundamental results on Haken 3-orbifolds 132 8.3. Kleinian groups 136 8.4. Thurston's gluing theorem 140 8.5. Thurston's Fixed Point Theorem 144 8.6. Thurston's Mirror Trick 147 9. Examples 159 9.1. A Euclidean collapse at angle 160 9.2. Another Euclidean collapse before TT 160 9.3. A Euclidean collapse at angle TT 161 9.4. Collapses at angle TT for Seifert fibred geometries 162 9.5. A collapse at n for Sol geometry 164 9.6. An essential Euclidean turnover : opening of a cusp 166 9.7. A pillow 169 9.8. An incompressible RP2(7r,7r) 171 A. Limit of Hyperbolicity for Spherical 3-Orbifolds, by Michael Heusener and Joan Porti 173 A.l. Proof of the main proposition 175 B. Thurston's hyperbolic Dehn filling Theorem 179 B.l. The manifold case 179 B.2. The orbifold case 187 B.3. Dehn filling with totally geodesic turnovers on the boundary 194 Bibliography 197 Index 207 ASTÉRISQUE 272 INTRODUCTION A 3-dimensional orbifold is a metrizable space with coherent local models given by quotients of R3 by finite subgroups of 0(3). For example, the quotient of a 3-manifold by a properly discontinuous group action naturally inherits a structure of a 3-orbifold. When the group action is finite, such an orbifold is said to be very good. For a general background about orbifolds see [BS1], [BS2], [DaM], [Kap, Chap. 7], [Sc3], [Takl] and [Thul, Chap. 13]. The purpose of this monograph is to give a complete proof of Thurston's orbifold theorem in the case where all local isotropy groups are cyclic subgroups of SO(3). Following [DaM], we say that such an orbifold is of cyclic type when in addition the ramification locus is non-empty. Hence a 3-orbifold O is of cyclic type iff its ramification locus £ is a non-empty 1-dimensional submanifold of the underlying manifold which is transverse to the boundary d\0\ = \dO\. The first result presented here is the following version of Thurston's Orbifold Theorem: Theorem 1. — Let O be a compact, connected, orientable, irreducible, and d-incom­ pressible 3-orbifold of cyclic type. If O is very good, topologically atoroidal and acylin- drical, then O is geometric (i. e. O admits either a hyperbolic, a Euclidean, or a Seifert fibred structure). Remark. — When dO is a union of toric 2-suborbifolds, the hypothesis that O is acylindrical is not needed. If dO 7^ 0 and O is not /-fibred, then O admits a hyperbolic structure of finite volume with totally geodesic boundary and cusps. We only consider smooth orbifolds, so that the local isotropy groups are always orthogonal. We recall that an orbifold is said to be good if it has a covering which is a manifold. Moreover if this covering is finite then the orbifold is said to be very good. 2 INTRODUCTION A general compact orientable irreducible and atoroidal 3-orbifold (which is not a priori very good) can be canonically split along a maximal (perhaps empty) collection of disjoint and non-parallel hyperbolic turnovers (i.e. a 2-orbifold with underlying space a 2-sphere and with three branching points) into either small or Haken 3- suborbifolds. An orientable compact 3-orbifold O is small if it is irreducible, its boundary dO is a (perhaps empty) collection of turnovers, and O does not contain any essential orientable 2-suborbifold. Using Theorem 1, we are able to geometrize such small 3-orbifolds, and hence to show that they are in fact very good. Theorem 2. — Let O be a compact, orientable, connected, small 3-orbifold of cyclic type. Then O is geometric. Therefore, to get a complete picture (avoiding the very good hypothesis), it remains to geometrize the Haken atoroidal pieces. An orientable compact 3-orbifold O is Haken if: - Ö is irreducible, - every embedded turnover is parallel to the boundary - and O contains an embedded orientable incompressible 2-suborbifold different from a turnover. The geometrization of Haken atoroidal 3-orbifolds relies on the following extension of Thurston's hyperbolization theorem (for Haken 3-manifolds): Theorem 3 (Thurston's hyperbolization theorem). — Let Ö be a compact, orientable, connected, irreducible, Haken 3-orbifold. If O is topologically atoroidal and not Seifert fibred, nor Euclidean, then Ö is hyperbolic. It is a result of W. Dunbar [Dun2] that an orientable Haken 3-orbifold can be decomposed into either discal 3-orbifolds or thick turnovers (i.e. {turnovers} x[0,1]) by repeated cutting along 2-sided properly embedded essential 2-suborbifolds. Due to this fact, the proof of Theorem 3 follows exactly the scheme of the proof for Haken 3-manifolds [Thu2, Thu3, Thu5], [McMl], [Kap], [Otl, Ot2]. We do not give a detailed proof of it here, but we only present the main steps to take in consideration and indicate shortly how to handle them in Chapter 8.