A rigidity result asserting that for complete finite-volume hyperbolic manifolds of dimension at least three, the geometric structure is uniquely determined by the fundamental group: any isomorphism between fundamental groups is realized by an isometry between the manifolds (hence geometry is determined by the group).

Principle

The organizing principle is that in dimensions ≥3 negative-curvature homogeneous geometries are rigid: discrete groups of isometries of hyperbolic space have no nontrivial deformations that change the quotient's hyperbolic structure, so algebraic data (π1) fixes geometric data.

Demonstration

Concrete statement: if M and N are closed hyperbolic n-manifolds (n≥3) and φ: π1(M)→π1(N) is an isomorphism, then there exists a unique isometry f: M→N inducing φ on fundamental groups. As a consequence, volume and lengths of closed geodesics are topological invariants.

Misapplication

Applying Mostow rigidity in dimension two where Teichmüller theory produces nontrivial deformations, or assuming the theorem holds for variable negative curvature or for geometries other than real hyperbolic space without verifying hypotheses.

Consequence

Topological invariants (e.g., isomorphism class of π1) determine geometric invariants (metric up to isometry), implying, for instance, that volume is a topological invariant and eliminating continuous moduli of hyperbolic structures in dimensions ≥3.

Reversal

Reversal highlights the low-dimensional contrast: in dimension two the map from topology to geometry is not injective — surfaces of the same topological type admit continuous families of non-isometric hyperbolic metrics (Teichmüller space), exhibiting flexibility rather than rigidity.

Boundary

Applies to complete hyperbolic manifolds of finite volume in real dimension ≥3 (with Mostow–Prasad extending to finite-volume noncompact cases); it excludes dimension 2, variable curvature metrics, and other locally symmetric spaces unless hypotheses are adapted.

Semantic Tension

Tension between rigidity (Mostow) and flexibility (Teichmüller): both concern the relation between discrete group data and geometric structures, but yield opposite behaviors depending on dimension and geometric context.

Synthesis

Mostow rigidity states that in real hyperbolic geometry of dimension three and higher the algebraic datum of the fundamental group completely prescribes the geometric metric structure of the manifold, removing continuous deformations and making geometry a topological invariant.