Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by and extended to finite volume manifolds by in 3 dimensions, and by in all dimensions at least 3. gave an alternate proof using the Gromov norm. gave the simplest available proof. While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic -manifold (for ) is a point, for a hyperbolic surface of genus there is a moduli space of dimension that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory. There is also a rich theory of deformation spaces of hyperbolic structures on infinite volume manifolds in three dimensions. 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). Let be the -dimensional hyperbolic space. A complete hyperbolic manifold can be defined as a quotient of 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 and are complete finite-volume hyperbolic manifolds of dimension . If there exists an isomorphism then it is induced by a unique isometry from to . Here is the fundamental group of a manifold . If is an hyperbolic manifold obtained as the quotient of by a group then . An equivalent statement is that any homotopy equivalence from to can be homotoped to a unique isometry. The proof actually shows that if has greater dimension than then there can be no homotopy equivalence between them.