nLab good orbifold

Definition

An orbifold $\mathcal{X}$ is called good (Thurston 92, Ch. 13, Def. 13.2.3) or developable if it is isomorphic to a global quotient of a smooth manifold $M$ by the action of a discrete group $\flat G$ (not necessarily finite):

$$\mathcal{X} \,\simeq\, M \sslash \flat G$$

Otherwise $\mathcal{X}$ is called bad.

If $\mathcal{X}$ is even the global quotient of a smooth manifold by a finite group action one says that it is very good.

In the other direction, an orbifold that is the global quotient of a smooth manifold by some (compact) Lie group is called a presentable orbifold.

References

• William Thurston, Three-dimensional geometry and topology, preliminary draft, University of Minnesota, Minnesota, (1992). 1979 version, scan of 1991 copy (note the pages are upside down and need to be rotated after downloading), pdf of 2002 version.

The first three chapters are published in expanded form as

• The Geometry and Topology of Three-Manifolds (1997) Princeton University Press page

The book has its own Wikipedia page.