The theorem can be given in a geometric formulation (pertaining to finite-volume, complete manifolds), and in an algebraic formulation (pertaining to lattices in
Lie groups).
Geometric form Let \mathbb H^n be the n-dimensional
hyperbolic space. A complete hyperbolic manifold can be defined as a quotient of \mathbb H^n by a group of isometries acting freely and
properly discontinuously (it is equivalent to define it as a
Riemannian manifold with sectional curvature -1 which is
complete). It is of finite volume if the integral of a
volume form is finite (which is the case, for example, if it is compact). The Mostow rigidity theorem may be stated as: :
Suppose M and N are complete finite-volume hyperbolic manifolds of dimension n \ge 3. If there exists an isomorphism f\colon \pi_1(M) \to \pi_1(N) then it is induced by a unique isometry from M to N. Here \pi_1(X) is the
fundamental group of a manifold X. If X is a hyperbolic manifold obtained as the quotient of \mathbb H^n by a group \Gamma then \pi_1(X) \cong \Gamma. An equivalent statement is that any
homotopy equivalence from M to N can be homotoped to a unique isometry. The proof actually shows that if N has greater dimension than M then there can be no homotopy equivalence between them.
Algebraic form The group of isometries of hyperbolic space \mathbb H^n can be identified with the Lie group \mathrm{PO}(n,1) (the
projective orthogonal group of a
quadratic form of signature (n,1). Then the following statement is equivalent to the one above. :
Let n \ge 3 and \Gamma and \Lambda be two lattices in \mathrm{PO}(n,1) and suppose that there is a group isomorphism f\colon \Gamma \to \Lambda. Then \Gamma and \Lambda are conjugate in \mathrm{PO}(n,1). That is, there exists a g \in \mathrm{PO}(n,1) such that \Lambda = g \Gamma g^{-1}. In greater generality Mostow rigidity holds (in its geometric formulation) more generally for fundamental groups of all complete, finite volume, non-positively curved (without Euclidean factors)
locally symmetric spaces of dimension at least three, or in its algebraic formulation for all lattices in
simple Lie groups not locally isomorphic to \mathrm{SL}_2(\R). ==Applications==