Three-dimensional orbifolds and cone-manifolds.

*(English)*Zbl 0955.57014
MSJ Memoirs. 5. Tokyo: Mathematical Society of Japan (MSJ). ix, 170 p. (2000).

The main goal of this memoir is to give an introduction to the statement and basic ideas in the proof of the Orbifold Theorem due to Thurston [see, for example, W. P. Thurston, Three-dimensional geometry and topology, Princeton Math. Ser. 35 (1997; Zbl 0873.57001)]. The content of the book was presented in a series of fifteen lectures given by the authors at a Regional Workshop in Tokyo (July, 1998). The authors develop the basic properties of orbifolds and cone-manifolds, give an outline of a proof of the orbifold theorem, and discuss the Gromov theory of limits of metric spaces and the deformations of hyperbolic structures. This memoir is really a wonderful book containing many new ideas written by three of the most creative researchers in three-dimensional hyperbolic geometry. So I recommend the reading to any graduate student and researcher in topology and the geometry of 3-manifolds.

Reviewer: Alberto Cavicchioli (Modena)

##### MSC:

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

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

57N10 | Topology of general \(3\)-manifolds (MSC2010) |