Hyperbolic three-manifold ($M={\mathbb{H}}^3/\Gamma$)

A complete hyperbolic three-manifold is a complete Riemannian manifold of dimension three with constant sectional curvature $-1$. Its universal cover is hyperbolic space ${\mathbb{H}}^3$ and its deck group acts freely and discretely by Riemannian isometries. In the orientable case it is a torsion-free Kleinian group.

For complete finite-volume hyperbolic space manifolds of dimension at least three, an isomorphism of fundamental groups is induced, up to the usual basepoint-path choice, by a unique hyperbolic isometry. The finite-volume noncompact case includes the Prasad extension. Thus complete hyperbolic metrics on a fixed finite-volume three-manifold have no deformation moduli, unlike metrics on closed hyperbolic surfaces.