Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti
type: Item
instance of: scholarly article
zbMATH Open document ID: 0971.57004
author: Michel Boileau
author: Joan Porti
publication date: 2001-08-16T00:00:00Z
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 1634458
zbMATH Keywords: 3-orbifold
zbMATH Keywords: geometric structure
zbMATH Keywords: orbifold geometrization
MaRDI profile type: Publication
title: Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti
published in: Astérisque
review text: 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 NEWLINENEWLINENEWLINETheorem 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). NEWLINENEWLINENEWLINEHere 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, …
reviewed by: Bruno Zimmermann
description: scientific article; zbMATH DE number 1634458