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.
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Mostow rigidity theorem is a fundamental result in the field of differential geometry, particularly in the study of hyperbolic geometry. It states that if two closed manifolds (or more generally, two complete Riemannian manifolds that are simply connected and have constant negative curvature) are isometric to each other, then they are also equivalent up to a unique way of deforming them.