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 hyperbolic structures on a finite volume hyperbolic -manifold is a point, for a hyperbolic surface of genus there is a moduli space of dimension that parameterizes all metrics of constant curvature, 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, and in an algebraic formulation.

Geometric form

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 of finite volume if its volume is finite. The Mostow rigidity theorem may be stated as:
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.

Algebraic form

The group of isometries of hyperbolic space can be identified with the Lie group (the projective orthogonal group of a quadratic form of signature. Then the following statement is equivalent to the one above.

In greater generality

Mostow rigidity holds more generally for fundamental groups of all complete, finite volume locally symmetric spaces of dimension at least 3, or in its algebraic formulation for all lattices in simple Lie groups not locally isomorphic to.

Applications

It follows from the Mostow rigidity theorem that the group of isometries of a finite-volume hyperbolic n-manifold M is finite and isomorphic to.
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.