Geometrization of 3-dimensional orbifolds

@article{Boileau2000GeometrizationO3,
  title={Geometrization of 3-dimensional orbifolds},
  author={Michel Boileau and Bernhard Leeb and Joan Porti},
  journal={Annals of Mathematics},
  year={2000},
  volume={162},
  pages={195-290}
}
This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramification locus, then O is geometric (i.e. has a metric of constant curvature or is Seifert fibred). As a corollary, any smooth orientationpreserving nonfree finite group action on S3 is conjugate to an orthogonal action. 

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References

SHOWING 1-10 OF 109 REFERENCES
Geometrization of 3-orbifolds of cyclic type
We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is
Hierarchies for 3-orbifolds
We generalize to the category of orbifolds (topological spaces locally modelled on Euclidean space modulo a finite group) some fundamental theorems in the study of 3-manifolds, including the fact
Manifold covers of 3-orbifolds with geometric pieces
Abstract It is conjectured that any compact 3-orbifold containing no bad 2-suborbifolds is built from pieces whose interiors admit geometric structures. We prove that any 3-orbifold having such
The geometric realizations of the decompositions of 3-orbifold fundamental groups
Abstract We introduce a type of generalized orbifold called an “orbifold composition”. We study their topology and the extensions and deformations of the maps between them. As the main goal, we
Uniformization of small 3-orbifolds
Abstract We prove a uniformization theorem for small compact orientable 3-orbifolds, that implies Thurston's orbifold theorem.
Surface groups and 3-manifolds which fiber over the circle
Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose
Deformations of hyperbolic 3-cone-manifolds
We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most \pi can be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the
Renormalization and 3-Manifolds Which Fiber over the Circle
Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of
Regenerating hyperbolic and spherical cone structures from Euclidean ones
Abstract We show that, in some cases, a Euclidean cone structure on a closed 3-manifold can be deformed into hyperbolic or spherical cone structures by moving the singular angle. We describe other
The moduli space of hyperbolic cone structures
Let $\Sigma$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, \Sigma)$
...
1
2
3
4
5
...