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 (complete) hyperbolic structures on a finite volume hyperbolic
n
n>2
g>1
6g-6
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).
Let
Hn
n
Hn
Suppose
M
N
n\ge3
f\colon\pi1(M)\to\pi1(N)
M
N
Here
\pi1(X)
X
X
Hn
\Gamma
\pi1(X)\cong\Gamma
An equivalent statement is that any homotopy equivalence from
M
N
N
M
The group of isometries of hyperbolic space
Hn
PO(n,1)
(n,1)
Let
n\ge3
\Gamma
Λ
PO(n,1)
f\colon\Gamma\toΛ
\Gamma
Λ
PO(n,1)
g\inPO(n,1)
Λ=g\Gammag-1
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
SL2(\R)
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
\operatorname{Out}(\pi1(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 exist hyperbolic groups which are quasi-isometric but not commensurable to each other.