# 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.

## 147 Citations

Geometrization of Three-Dimensional Orbifolds via Ricci Flow

- Mathematics
- 2011

A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman along with earlier work of Boileau-Leeb-Porti and…

Weak collapsing and geometrisation of aspherical 3-manifolds

- Mathematics
- 2008

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick…

On the topology of locally volume collapsed Riemannian 3-orbifolds

- Mathematics
- 2011

We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad…

Small volume link orbifolds

- Mathematics
- 2013

This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the…

Complexity of 3-orbifolds☆

- Mathematics
- 2004

Abstract We extend Matveev's theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible…

Collapsing irreducible 3-manifolds with nontrivial fundamental group

- Mathematics
- 2009

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally…

Branchfold and rational conifolds

- Mathematics
- 2008

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings.…

Seiberg-Witten invariants of 3-orbifolds and non-Kähler surfaces

- Mathematics
- 2011

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant…

Regenerating hyperbolic cone structures from Nil

- Mathematics
- 2002

Let O be a three-dimensional Nil{orbifold, with branching locus a knot transverse to the Seifert bration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in ( "; ). We also…

A general theory of orbifolds, and essential 2-suborbifolds respecting graph products

- Mathematics
- 2016

Abstract We make an analogy of Culler–Morgan–Shalen theory for 3-orbifolds. Our main goal is to show that there exists a non-empty system of incompressible and not boundary parallel 2-suborbifolds…

## References

SHOWING 1-10 OF 109 REFERENCES

Geometrization of 3-orbifolds of cyclic type

- Mathematics
- 2001

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

- Mathematics
- 1988

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

- Mathematics
- 1989

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

- Mathematics
- 1999

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

- Mathematics
- 2001

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

- Physics, Mathematics
- 1980

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

- Mathematics
- 1998

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

- Mathematics
- 1996

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

- Mathematics
- 1998

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

- Mathematics
- 1998

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 $(X, \Sigma)$…