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.
Besson, Courtois & Gallot (1996) gave the simplest available proof.
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic
) is a point, for a hyperbolic surface of genus
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).
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).
The Mostow rigidity theorem may be stated as: Here
is an hyperbolic manifold obtained as the quotient of
The group of isometries of hyperbolic space
(the projective orthogonal group of a quadratic form of signature
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 not locally isomorphic to
It follows from the Mostow rigidity theorem that the group of isometries of a finite-volume hyperbolic n-manifold M (for n>2) is finite and isomorphic to
Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations of triangulated planar graphs.
[1] 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.