# Geometrization of 3-dimensional orbifolds

### Abstract

This paper is devoted to the proof of the orbifold theorem: If $\mathcal{O}$ is a compact connected orientable irreducible and topologically atoroidal $3$-orbifold with nonempty ramification locus, then $\mathcal{O}$ is geometric (i.e. has a metric of constant curvature or is Seifert fibred). As a corollary, any smooth orientation-preserving nonfree finite group action on $S^3$ is conjugate to an orthogonal action.

## Authors

Michel Boileau

Laboratoire Émile Picard, CNRS UMR 5580, Université Paul Sabatier, 31062 Toulouse, France

Bernhard Leeb

Mathematisches Institut, Universität München, 80333 München, Germany

Joan Porti

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain