# Mostow rigidity theorem

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In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a 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 dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm.

Weil (1960, 1962) proved a closely related theorem, that implies in particular that cocompact discrete groups of isometries of hyperbolic space of dimension at least 3 have no non-trivial deformations.

While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic 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. In dimension three, there is a "non-rigidity" theorem due to Thurston called the hyperbolic Dehn surgery theorem; it allows one to deform hyperbolic structures on a finite volume manifold as long as changing homeomorphism type is allowed. In addition, there is a rich theory of deformation spaces of hyperbolic structures on infinite volume manifolds.

## The theorem

The theorem can be given in a geometric formulation, and in an algebraic formulation.

### Geometric form

The Mostow rigidity theorem may be stated as:

Suppose M and N are complete finite-volume hyperbolic n-manifolds with n > 2. If there exists an isomorphism ƒ : π1(M) → π1(N) then it is induced by a unique isometry from M to N.

Here, π1(M) is the fundamental group of a manifold M.

Another version is to state 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

An equivalent formulation is:

Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n-space H with n > 2 whose quotients Hand Hhave finite volume. If Γ and Δ are isomorphic as discrete groups, then they are conjugate.

## Applications

The group of isometries of a finite-volume hyperbolic n-manifoldM (for n>2) is finite and isomorphic to Out(π1(M)).

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