Mostow rigidity

Mostow rigidity is the theorem, proved by George Mostow in 1968, that the geometry of a closed hyperbolic manifold of dimension at least three is completely determined by its fundamental group. Unlike surfaces, where hyperbolic structures vary in Teichmüller space, higher-dimensional hyperbolic manifolds are rigid: any isomorphism of fundamental groups is induced by a unique isometry. This rigidity creates a profound bridge between the algebraic world of group theory and the geometric world of hyperbolic geometry, and it underpins the uniqueness claims in the geometrization program by ensuring that when a 3-manifold is hyperbolic, its geometry is not merely one option among many but the only option.