Mostow rigidity theorem

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

References

• Gromov, Michael (1981), "Hyperbolic manifolds (according to Thurston and Jørgensen)", Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., 842, Berlin, New York: Springer-Verlag, pp. 40–53, doi:10.1007/BFb0089927, ISBN 978-3-540-10292-2, MR636516, http://www.numdam.org/numdam-bin/fitem?id=SB_1979-1980__22__40_0 
• Marden, Albert (1974), "The geometry of finitely generated kleinian groups", Annals of Mathematics. Second Series 99: 383–462, ISSN 0003-486X, JSTOR 1971059, MR0349992, Zbl 0282.30014 
• G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms, Publ. Math. IHES 34 (1968) 53–104.
• Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals of mathematics studies, 78, Princeton University Press, ISBN 978-0-691-08136-6, MR0385004, http://books.google.com/books?id=xT0SFmrFrWoC 
• Prasad, Gopal (1973), "Strong rigidity of Q-rank 1 lattices", Inventiones Mathematicae 21: 255–286, doi:10.1007/BF01418789, ISSN 0020-9910, MR0385005 
• 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)
• Weil, André (1960), "On discrete subgroups of Lie groups", Annals of Mathematics. Second Series 72: 369–384, ISSN 0003-486X, JSTOR 1970140, MR0137792 
• Weil, André (1962), "On discrete subgroups of Lie groups. II", Annals of Mathematics. Second Series 75: 578–602, ISSN 0003-486X, JSTOR 1970212, MR0137793