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

The notion of good orbifolds is due to Def. 13.2.3 in:

• William Thurston: Three-dimensional geometry and topology, preliminary draft, University of Minnesota (1992) [1979: ark:/13960/t3714t34v, 1991: pdf, 2002: pdf, pdf]

  the first three chapters of which are published in expanded form as:

• William Thurston: The Geometry and Topology of Three-Manifolds, Princeton University Press (1997) [ISBN:9780691083049, Wikipedia page]

Further discussion:

• Christian Lange: Good, but not very good orbifolds [arXiv:2404.14234]