The Geometrization ConjectureThis book gives a complete proof of the geometrization conjecture, which describes all compact 3manifolds in terms of geometric pieces, i.e., 3manifolds with locally homogeneous metrics of finite volume. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the GromovHausdorff limits of sequences of more and more locally volume collapsed 3manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces. In the course of proving the geometrization conjecture, the authors provide an overview of the main results about Ricci flows with surgery on 3dimensional manifolds, introducing the reader to this difficult material. The book also includes an elementary introduction to GromovHausdorff limits and to the basics of the theory of Alexandrov spaces. In addition, a complete picture of the local structure of Alexandrov surfaces is developed. All of these important topics are of independent interest. Titles in this series are copublished with the Clay Mathematics Institute (Cambridge, MA).

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Contents
Introduction  1 
GEOMETRIC AND ANALYTIC RESULTS  11 
Limits as t  25 
Local results valid for large time  41 
Proofs of the three propositions  66 
Part 2  95 
The collapsing theorem  101 
Overview of the rest of the argument  107 
3dimensional analogues  163 
LOCALLY VOLUME COLLAPSED 3MANIFOLDS  181 
13  205 
14  226 
The equivariant case  245 
Bibliography  281 
21  282 
289  