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.

) is a point, for a hyperbolic surface of genus there is a moduli space 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

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

-dimensional hyperbolic space 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:

1. . If there exists an isomorphism.

is the fundamental group.

An equivalent statement is that any homotopy equivalence then there can be no homotopy equivalence between them.

Algebraic form

(the projective orthogonal group of a quadratic form of signature. Then the following statement is equivalent to the one above.

1. be two lattices.

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.

Applications

.

Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations of triangulated planar graphs.

A consequence of Mostow rigidity of interest in geometric group theory is that there exist hyperbolic groups which are quasi-isometric but not commensurable to each other.

See also

• Superrigidity, a stronger result for higher-rank spaces

• Local rigidity, a result about deformations that are not necessarily lattices.

Notes

References

• . (Provides a survey of a large variety of rigidity theorems, including those concerning Lie groups, algebraic groups and dynamics of flows. Includes 230 references.)

• . (Gives two proofs: one similar to Mostow's original proof, and another based on the Gromov norm)