Mostow rigidity

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 Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm.

While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic ${\displaystyle n}$-manifold (for n > 2) is a point, for a hyperbolic surface of genus g > 1 there is a moduli space of dimension 6g − 6 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 ${\displaystyle \mathbb {H} ^{n}}$ be the ${\displaystyle n}$-dimensional hyperbolic space. A complete hyperbolic manifold can be defined as a quotient of ${\displaystyle \mathbb {H} ^{n}}$ 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 its volume is finite (for example if it is compact). The Mostow rigidity theorem may be stated as:

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