Hyperbolic three-orbifold

A hyperbolic three-orbifold is a three-dimensional orbifold with local models $U/G$ , where $U$ is open in hyperbolic three-space of sectional curvature $-1$ , the finite group $G$ acts by isometries, and chart embeddings are local isometries.

Complete quotients

A Kleinian group $\Gamma$ gives a complete orientable example $\Gamma\backslash\mathbb H^3$ . Properness of the discrete isometric action provides local charts using the finite stabilizers of points.

When the action is free, the quotient is a hyperbolic manifold. A nontrivial finite stabilizer produces orbifold isotropy, even if the underlying quotient happens to be a topological manifold.

Volume

Use the Riemannian volume at curvature $-1$ . In a finite quotient chart, integrating an invariant density downstairs means dividing the upstairs integral by the group order. This convention is the one used for the Bianchi volume.

References

A. Adem and M. Klaus, Lectures on orbifolds and group cohomology, §2, Definitions 2.1–2.2. Author-hosted notes See also its quotient-orbifold construction.