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.
Contents
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic
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
Let
Here
An equivalent statement is that any homotopy equivalence from
Algebraic form
The group of isometries of hyperbolic space
In greater generality
Mostow rigidity holds (in its geometric formulation) 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 (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.
A consequence of Mostow rigidity of interest in geometric group theory is that there exists hyperbolic groups which are quasi-isometric but not commensurable to each other.