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# Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group and hence unique. It is the leading example of the types of statements that occur in rigidity theory.

While the theorem shows that there is no deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n-manifold (for n >2), 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 was proven for the closed case by G. D. Mostow in 1968 and extended to the finite volume case by G. Prasad (and independently Marden). Sometimes the theorem is called the Mostow-Prasad rigidity theorem. This theorem has been generalized to non-uniform lattices. An important alternate proof using the Gromov norm was given by M. Gromov in 1979.

## 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 $f:pi_1\left(M\right)topi_1\left(N\right)$ then it is induced by a unique isometry from M to N.

Here, $pi_1\left(M\right)$ is the fundamental group of the manifold.

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 H/Γ and H/Δ have finite volume. If they are isomorphic, then they are conjugate.

## Applications

An important corollary is that a finite volume hyperbolic n-manifold M for n > 2 has no nontrivial inner automorphisms of $pi_1\left(M\right)$. One can conclude that the group of isometries of M is finite and isomorphic to $Out\left(pi_1\left(M\right)\right)$.

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

## References

• G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms, Publ. Math. IHES 34 (1968) 53-104.
• M. Gromov, Hyperbolic manifolds according to Thurston and Jorgensen, Séminaire Bourbaki, 32eme année, 1979/80, pp 40-53.
• R. J. Spatzier, Harmonic Analysis in Rigidity Theory, (1993) pp. 153-205, appearing in Ergodic Theory and its Connection with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, Karl. E. Petersen, Ibrahim A. Salama, eds. Cambridge University Press (1995) ISBN 0-521-45999-0. (Provides a survey of a large variety of rigidity theorems, including those concerning Lie groups, algebraic groups and dynamics of flows. Includes 230 references.)
• William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981). (Gives two proofs: one similar to Mostow's original proof, and another based on the Gromov norm)

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