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Three-dimensional orbifolds and their geometric structures. (English) Zbl 1058.57009
Panoramas et Synthèses 15. Paris: Société Mathématique de France (ISBN 2-85629-152-X ). viii, 167 p. (2003).
The Orbifold Theorem asserts that if \(\mathcal{O}\) is a compact, connected, orientable, irreducible, atoroidal \(3\)-orbifold with non-empty singular locus, then \(\mathcal{O}\) has a geometric structure (in the sense of orbifolds). This result was first announced by Thurston in 1981, but because of its formidable technical difficulties, substantial progress on a complete written proof has only been made in recent years. As of this writing, the most fully-developed proofs are the works of M. Boileau, B. Leeb and J. Porti [Geometrization of \(3\)-dimensional orbifolds, Mathematics ArXiv GT/0010184], and [Geometrization of 3-dimensional orbifolds, Part I: Geometry of cone manifolds, Mathematics ArXiv GT/0010185], M. Boileau and J. Porti [Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti., (Astérisque. 272. Paris: Société Mathématique de France.) (2001; Zbl 0971.57004)], and the work of D. Cooper, C. D. Hodgson and S. P. Kerckhoff [Three-dimensional orbifolds and cone-manifolds, (Math. Soc. Japan, Tokyo), (2000; Zbl 0955.57014)].
This well-written monograph presents the extensive background theory and results needed for the Orbifold Theorem, and finishes with a detailed outline of the proof in the important special case when the orbifold is closed and all local groups are cyclic. The background includes Thurston’s eight geometries, the topological theory of \(2\)- and \(3\)-dimensional orbifolds, fibered structures on \(3\)-orbifolds, factorization of \(3\)-orbifolds into irreducible orbifolds, Haken orbifolds and their toric splittings, geometrization of Seifert orbifolds, hyperbolic structures and the associated representation varieties, hyperbolic Dehn filling of orbifolds, and cone manifolds and their deformation theory. Some of these are straightforward adaptations of corresponding concepts from the theory of \(3\)-manifolds, but many require significant technical innovations which have been developed by the authors and others. Extensive citations are included; the bibliography lists well over two hundred references.
The proof outlined in the final chapter of the monograph is for the case of orbifolds of cyclic type, that is, when all local isotropy groups are cyclic. Here is a very superficial sketch. The orbifold is first split into canonical pieces that are Haken or small. The Haken pieces can be geometrized by adapting Thurston’s geometrization theorem for \(3\)-manifolds to the orbifold case. For the small pieces that are not of the easily-handled Seifert-fibered case, there is a hyperbolic structure on the complement of the singular locus. One considers a one-parameter family of cone manifolds with singular set the singular locus of the orbifold, where the parameter value \(t=0\) corresponds to the complete hyperbolic structure on the complement and \(t=1\) corresponds to the original orbifold. Supposing that \(t=1\) does not correspond to a hyperbolic cone-structure, the proof examines the situation at the infimum of the non-hyperbolic-cone-structure parameters, producing a geometric structure in various different ways depending on the nature of the singular behavior at that parameter.
After such a careful and thorough presentation of the ingredients and the proof of this special case, some discussion of the additional technical difficulties encountered the general case could perhaps have been included. This quibble aside, the authors have provided a useful reference for this major work.

57M50 General geometric structures on low-dimensional manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M60 Group actions on manifolds and cell complexes in low dimensions
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes