A classical result in two dimensions says that a compact surface is diffeomorphic to the quotient of one of three model geometries (the sphere, the Euclidean plane and the hyperbolic plane) by a discrete subgroup of the group of all isometries. In three dimensions, there are eight model geometries. The Thurston geometrization conjecture claims, roughly speaking, that any compact three-dimensional manifold is uniquely decomposable along a finite set of embedded surfaces into quotients of model three-dimensional geometries. It results in a few other important conjectures (including the famous Poincaré conjecture). Studying quotients of manifolds by discrete subgroups, it is natural to work in a broader category of orbifolds. The main aim of the book is to describe this circle of ideas. The authors discuss in turn homogeneous 3-dimensional geometries, canonical decompositions, Haken orbifolds, Seifert fibered orbifolds, the Thurston hyperbolization theorem, varieties of representations, hyperbolic Dehn filling for orbifolds and the orbifold theorem. The Perelman results based on different techniques (the Ricci flow equation) are not included. The book is well organized and nicely written.

Reviewer:

vs