##
**Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti.**
*(English)*
Zbl 0971.57004

Astérisque. 272. Paris: Société Mathématique de France. vi, 208 p. (2001).

Thurston’s geometrization conjecture in 3-dimensional topology states that every compact orientable irreducible 3-orbifold (including 3-manifolds) has a canonical decomposition along Euclidean 2-suborbifolds into geometric 3-orbifolds (mainly hyperbolic and Seifert fibered 3-orbifolds). For Haken 3-manifolds this follows from Thurston’s manifold hyperbolization theorem and the Jaco-Shalen-Johannson decomposition. The main results of the present monograph show that the conjecture holds for (very good) 3-orbifolds of cyclic type, i.e. with nonempty singular set consisting of circles and intervals (in general, the singular set of a compact orientable 3-orbifold is a 3-valent graph). In more technical terms, the first main result of the monograph is the following

Theorem 1: Let \(\mathcal O\) be a compact connected orientable irreducible 3-orbifold of cyclic type. If \(\mathcal O\) is very good, topologically atoroidal and acylindrical, then \(\mathcal O\) is geometric (hyperbolic, Euclidean or Seifert fibered).

Here a 3-orbifold is called very good if it admits a finite covering by a 3-manifold. This hypothesis allows to apply the variety of known results about 3-manifolds (or their equivariant versions) and is presently mainly due to the technical structure of the proof. Theorem 1 applies, for example, to any 3-orbifold which is the 3-sphere with a hyperbolic knot as singular set, of singularity index \(p>2\); in particular, it follows that the \(p\)-fold cyclic branched covering of a hyperbolic knot is a hyperbolic 3-manifold, with the only exception of the 3-fold cyclic branched covering of the figure-8 knot which is Euclidean. However, in general it is not easy to prove that an orbifold is very good so one would like to get rid of this hypothesis. In the present monograph, this is achieved by distinguishing two complementary classes of atoroidal 3-orbifolds: small orbifolds and Haken 3-orbifolds. For small orbifolds which, by definition, do not contain essential orientable 2-suborbifolds and whose boundary consists of “turnovers”, the methods of the proof of Theorem 1 apply to show that small 3-orbifolds of cyclic type are geometric, avoiding the hypothesis of being very good. For atoroidal Haken 3-orbifolds, a generalization of Thurston’s hyperbolization theorem for Haken 3-manifold is invoked, and the main steps of a proof are sketched. As any compact atoroidal 3-orbifold can be decomposed into small 3-orbifolds and Haken 3-orbifolds, these results combine to the orbifold geometrization theorem as announced by Thurston, for 3-orbifolds of cyclic type:

Let \(\mathcal O\) be a compact connected orientable irreducible 3-orbifold of cyclic type; if \(\mathcal O\) is topologically atoroidal, then \(\mathcal O\) is geometric. (Also, any compact irreducible 3-orbifold possesses a canonical decomposition into geometric 3-orbifolds; then, a posteriori, such orbifolds are also very good).

In two recent preprints the two authors, together with B. Leeb, proved the orbifold geometrization theorem for compact orientable 3-orbifolds with arbitrary singular sets, i.e. not necessarily of cyclic type. An alternative approach to the orbifold geometrization has been worked out by D. Cooper, C. D. Hodgson and St. P. Kerckhoff [Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5 (2000; Zbl 0955.57014)].

The main tools of the proofs in the present monograph are Thurston’s hyperbolization theorem for Haken 3-manifolds, the hyperbolic Dehn surgery (or Dehn filling) theorem and results about the geometric convergence of sequences of hyperbolic cone-manifolds. Following Thurston’s original approach, the structure of the proof of Theorem 1 is as follows. One reduces to the case that the complement of the singular set of the orbifold has a complete hyperbolic structure. The hyperbolic Dehn surgery theorem implies that, for small angles around the singular set, the corresponding cone manifolds have hyperbolic structures (in contrast to geometric 3-orbifolds whose angles around the singular set are submultiples of \(2\pi\), for a geometric cone manifold arbitrary angles are allowed). Then one tries to increase these cone angles in order to arrive at the cone angles of the original orbifold, studying the geometric limits and possible types of degeneration of the corresponding sequences of hyperbolic cone manifolds. This uses a cone manifold version of Gromov’s compactness theorem for Riemannian manifolds with pinched sectional curvature. In the nicest (non-collapsing) case the hyperbolic cone manifold structures converge to a hyperbolic structure on the original orbifold, in the collapsing case one has to study carefully the possible limits (possibly after rescaling the hyperbolic cone metrics) which may be Euclidean 3-orbifolds, Seifert orbifolds or may contain Euclidean 2-suborbifolds. These geometric convergence theorems are among the main results of the monograph, their proofs occupy chapters 3-6 and are the heart of the monograph. “The authors’ main contribution takes place in the analysis of the so-called collapsing case. There they use the notion of simplicial volume due to Gromov and a cone manifold version of his isolation theorem. This gives a simpler combinatorial approach to collapses than Thurston’s original one. In particular, it spares us the difficult task of establishing a suitable Cheeger-Gromov theory for collapses of cone manifolds.” In the last chapter (9), explicit examples of collapses of hyperbolic cone structures to other geometric structures are discussed. In two appendices, limits of hyperbolicity for spherical 3-orbifolds are considered (when deforming the hyperbolic structures of the complement of the singular set, these cannot converge directly to the spherical structure but one arrives at a Euclidean structure first), and a proof of Thurston’s hyperbolic Dehn surgery theorem is given.

Theorem 1: Let \(\mathcal O\) be a compact connected orientable irreducible 3-orbifold of cyclic type. If \(\mathcal O\) is very good, topologically atoroidal and acylindrical, then \(\mathcal O\) is geometric (hyperbolic, Euclidean or Seifert fibered).

Here a 3-orbifold is called very good if it admits a finite covering by a 3-manifold. This hypothesis allows to apply the variety of known results about 3-manifolds (or their equivariant versions) and is presently mainly due to the technical structure of the proof. Theorem 1 applies, for example, to any 3-orbifold which is the 3-sphere with a hyperbolic knot as singular set, of singularity index \(p>2\); in particular, it follows that the \(p\)-fold cyclic branched covering of a hyperbolic knot is a hyperbolic 3-manifold, with the only exception of the 3-fold cyclic branched covering of the figure-8 knot which is Euclidean. However, in general it is not easy to prove that an orbifold is very good so one would like to get rid of this hypothesis. In the present monograph, this is achieved by distinguishing two complementary classes of atoroidal 3-orbifolds: small orbifolds and Haken 3-orbifolds. For small orbifolds which, by definition, do not contain essential orientable 2-suborbifolds and whose boundary consists of “turnovers”, the methods of the proof of Theorem 1 apply to show that small 3-orbifolds of cyclic type are geometric, avoiding the hypothesis of being very good. For atoroidal Haken 3-orbifolds, a generalization of Thurston’s hyperbolization theorem for Haken 3-manifold is invoked, and the main steps of a proof are sketched. As any compact atoroidal 3-orbifold can be decomposed into small 3-orbifolds and Haken 3-orbifolds, these results combine to the orbifold geometrization theorem as announced by Thurston, for 3-orbifolds of cyclic type:

Let \(\mathcal O\) be a compact connected orientable irreducible 3-orbifold of cyclic type; if \(\mathcal O\) is topologically atoroidal, then \(\mathcal O\) is geometric. (Also, any compact irreducible 3-orbifold possesses a canonical decomposition into geometric 3-orbifolds; then, a posteriori, such orbifolds are also very good).

In two recent preprints the two authors, together with B. Leeb, proved the orbifold geometrization theorem for compact orientable 3-orbifolds with arbitrary singular sets, i.e. not necessarily of cyclic type. An alternative approach to the orbifold geometrization has been worked out by D. Cooper, C. D. Hodgson and St. P. Kerckhoff [Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5 (2000; Zbl 0955.57014)].

The main tools of the proofs in the present monograph are Thurston’s hyperbolization theorem for Haken 3-manifolds, the hyperbolic Dehn surgery (or Dehn filling) theorem and results about the geometric convergence of sequences of hyperbolic cone-manifolds. Following Thurston’s original approach, the structure of the proof of Theorem 1 is as follows. One reduces to the case that the complement of the singular set of the orbifold has a complete hyperbolic structure. The hyperbolic Dehn surgery theorem implies that, for small angles around the singular set, the corresponding cone manifolds have hyperbolic structures (in contrast to geometric 3-orbifolds whose angles around the singular set are submultiples of \(2\pi\), for a geometric cone manifold arbitrary angles are allowed). Then one tries to increase these cone angles in order to arrive at the cone angles of the original orbifold, studying the geometric limits and possible types of degeneration of the corresponding sequences of hyperbolic cone manifolds. This uses a cone manifold version of Gromov’s compactness theorem for Riemannian manifolds with pinched sectional curvature. In the nicest (non-collapsing) case the hyperbolic cone manifold structures converge to a hyperbolic structure on the original orbifold, in the collapsing case one has to study carefully the possible limits (possibly after rescaling the hyperbolic cone metrics) which may be Euclidean 3-orbifolds, Seifert orbifolds or may contain Euclidean 2-suborbifolds. These geometric convergence theorems are among the main results of the monograph, their proofs occupy chapters 3-6 and are the heart of the monograph. “The authors’ main contribution takes place in the analysis of the so-called collapsing case. There they use the notion of simplicial volume due to Gromov and a cone manifold version of his isolation theorem. This gives a simpler combinatorial approach to collapses than Thurston’s original one. In particular, it spares us the difficult task of establishing a suitable Cheeger-Gromov theory for collapses of cone manifolds.” In the last chapter (9), explicit examples of collapses of hyperbolic cone structures to other geometric structures are discussed. In two appendices, limits of hyperbolicity for spherical 3-orbifolds are considered (when deforming the hyperbolic structures of the complement of the singular set, these cannot converge directly to the spherical structure but one arrives at a Euclidean structure first), and a proof of Thurston’s hyperbolic Dehn surgery theorem is given.

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57M50 | General geometric structures on low-dimensional manifolds |

53C20 | Global Riemannian geometry, including pinching |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |