# Mostow rigidity theorem

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Theorem in hyperbolic geometry

In [mathematics](/source/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](/source/Hyperbolic_manifold) of dimension greater than two is determined by the [fundamental group](/source/Fundamental_group) and hence unique. The theorem was proven for [closed manifolds](/source/Closed_manifold) by [Mostow](/source/George_Mostow) ([1968](#CITEREFMostow1968)) and extended to finite volume manifolds by [Marden (1974)](#CITEREFMarden1974) in 3 dimensions, and by [Prasad](/source/Gopal_Prasad) ([1973](#CITEREFPrasad1973)) in all dimensions at least 3. [Gromov (1981)](#CITEREFGromov1981) gave an alternate proof using the [Gromov norm](/source/Gromov_norm). [Besson, Courtois & Gallot (1996)](#CITEREFBessonCourtoisGallot1996) gave the simplest available proof.

While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n {\displaystyle n} -manifold (for n > 2 {\displaystyle n>2} ) is a point, for a hyperbolic surface of [genus](/source/Genus_(mathematics)) g > 1 {\displaystyle g>1} there is a [moduli space](/source/Moduli_space) of dimension 6 g − 6 {\displaystyle 6g-6} that parameterizes all metrics of constant curvature (up to [diffeomorphism](/source/Diffeomorphism)), a fact essential for [Teichmüller theory](/source/Teichm%C3%BCller_theory). There is also a rich theory of deformation spaces of hyperbolic structures on *infinite* volume manifolds in three dimensions.

## 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](/source/Lie_group)).

### Geometric form

Let H n {\displaystyle \mathbb {H} ^{n}} be the n {\displaystyle n} -dimensional [hyperbolic space](/source/Hyperbolic_space). A complete hyperbolic manifold can be defined as a quotient of H n {\displaystyle \mathbb {H} ^{n}} by a group of isometries acting freely and [properly discontinuously](/source/Group_action_(mathematics)#Types_of_action) (it is equivalent to define it as a [Riemannian manifold with sectional curvature -1](/source/Hyperbolic_manifold) which is [complete](/source/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces)). It is of finite volume if the integral of a [volume form](/source/Volume_form) is finite (which is the case, for example, if it is compact). The Mostow rigidity theorem may be stated as:

- *Suppose M {\displaystyle M} and N {\displaystyle N} are complete finite-volume hyperbolic manifolds of dimension n ≥ 3 {\displaystyle n\geq 3} . If there exists an [isomorphism](/source/Isomorphism) f : π 1 ( M ) → π 1 ( N ) {\displaystyle f\colon \pi _{1}(M)\to \pi _{1}(N)} then it is induced by a unique isometry from M {\displaystyle M} to N {\displaystyle N} .*

Here π 1 ( X ) {\displaystyle \pi _{1}(X)} is the [fundamental group](/source/Fundamental_group) of a manifold X {\displaystyle X} . If X {\displaystyle X} is a hyperbolic manifold obtained as the quotient of H n {\displaystyle \mathbb {H} ^{n}} by a group Γ {\displaystyle \Gamma } then π 1 ( X ) ≅ Γ {\displaystyle \pi _{1}(X)\cong \Gamma } .

An equivalent statement is that any [homotopy equivalence](/source/Homotopy_equivalence) from M {\displaystyle M} to N {\displaystyle N} can be homotoped to a unique isometry. The proof actually shows that if N {\displaystyle N} has greater dimension than M {\displaystyle M} then there can be no homotopy equivalence between them.

### Algebraic form

The group of isometries of hyperbolic space H n {\displaystyle \mathbb {H} ^{n}} can be identified with the Lie group P O ( n , 1 ) {\displaystyle \mathrm {PO} (n,1)} (the [projective orthogonal group](/source/Projective_orthogonal_group) of a [quadratic form of signature](/source/Quadratic_form#Real_quadratic_forms) ( n , 1 ) {\displaystyle (n,1)} . Then the following statement is equivalent to the one above.

- *Let n ≥ 3 {\displaystyle n\geq 3} and Γ {\displaystyle \Gamma } and Λ {\displaystyle \Lambda } be two [lattices](/source/Lattice_(discrete_subgroup)) in P O ( n , 1 ) {\displaystyle \mathrm {PO} (n,1)} and suppose that there is a group isomorphism f : Γ → Λ {\displaystyle f\colon \Gamma \to \Lambda } . Then Γ {\displaystyle \Gamma } and Λ {\displaystyle \Lambda } are conjugate in P O ( n , 1 ) {\displaystyle \mathrm {PO} (n,1)} . That is, there exists a g ∈ P O ( n , 1 ) {\displaystyle g\in \mathrm {PO} (n,1)} such that Λ = g Γ g − 1 {\displaystyle \Lambda =g\Gamma g^{-1}} .*

### In greater generality

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](/source/Locally_symmetric_space) of dimension at least three, or in its algebraic formulation for all lattices in [simple Lie groups](/source/Simple_Lie_group) not locally isomorphic to S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} .

## 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 ⁡ ( π 1 ( M ) ) {\displaystyle \operatorname {Out} (\pi _{1}(M))} .

Mostow rigidity was also used by Thurston to prove the uniqueness of [circle packing representations](/source/Circle_packing_theorem) of [triangulated planar graphs](/source/Planar_graph).[1]

A consequence of Mostow rigidity of interest in [geometric group theory](/source/Geometric_group_theory) is that there exist [hyperbolic groups](/source/Hyperbolic_group) which are [quasi-isometric](/source/Quasi-isometry) but not [commensurable](/source/Commensurability_(group_theory)) to each other.

## See also

- [Superrigidity](/source/Superrigidity), a stronger result for higher-rank spaces

- [Local rigidity](/source/Local_rigidity), a result about deformations that are not necessarily lattices.

## Notes

1. **[^](#cite_ref-FOOTNOTEThurston1978–1981Chapter_13_1-0)** [Thurston 1978–1981](#CITEREFThurston1978–1981), Chapter 13.

## References

- Besson, Gérard; Courtois, Gilles; [Gallot, Sylvestre](/source/Sylvestre_Gallot) (1996), "Minimal entropy and Mostow's rigidity theorems", *Ergodic Theory and Dynamical Systems*, **16** (4): 623–649, [doi](/source/Doi_(identifier)):[10.1017/S0143385700009019](https://doi.org/10.1017%2FS0143385700009019), [S2CID](/source/S2CID_(identifier)) [122773907](https://api.semanticscholar.org/CorpusID:122773907)

- Gromov, Michael (1981), ["Hyperbolic manifolds (according to Thurston and Jørgensen)"](https://web.archive.org/web/20160110061753/http://www.numdam.org/numdam-bin/fitem?id=SB_1979-1980__22__40_0), [*Bourbaki Seminar, Vol. 1979/80*](https://www.numdam.org/article/SB_1979-1980__22__40_0.pdf) (PDF), Lecture Notes in Math., vol. 842, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), pp. 40–53, [doi](/source/Doi_(identifier)):[10.1007/BFb0089927](https://doi.org/10.1007%2FBFb0089927), [ISBN](/source/ISBN_(identifier)) [978-3-540-10292-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-10292-2), [MR](/source/MR_(identifier)) [0636516](https://mathscinet.ams.org/mathscinet-getitem?mr=0636516), archived from [the original](http://www.numdam.org/numdam-bin/fitem?id=SB_1979-1980__22__40_0) on 2016-01-10

- Marden, Albert (1974), "The geometry of finitely generated kleinian groups", *[Annals of Mathematics](/source/Annals_of_Mathematics)*, Second Series, **99** (3): 383–462, [doi](/source/Doi_(identifier)):[10.2307/1971059](https://doi.org/10.2307%2F1971059), [ISSN](/source/ISSN_(identifier)) [0003-486X](https://search.worldcat.org/issn/0003-486X), [JSTOR](/source/JSTOR_(identifier)) [1971059](https://www.jstor.org/stable/1971059), [MR](/source/MR_(identifier)) [0349992](https://mathscinet.ams.org/mathscinet-getitem?mr=0349992), [Zbl](/source/Zbl_(identifier)) [0282.30014](https://zbmath.org/?format=complete&q=an:0282.30014)

- Mostow, G. D. (1968), ["Quasi-conformal mappings in *n*-space and the rigidity of the hyperbolic space forms"](https://www.numdam.org/item?id=PMIHES_1968__34__53_0), *Publ. Math. IHÉS*, **34**: 53–104, [doi](/source/Doi_(identifier)):[10.1007/bf02684590](https://doi.org/10.1007%2Fbf02684590), [S2CID](/source/S2CID_(identifier)) [55916797](https://api.semanticscholar.org/CorpusID:55916797)

- Mostow, G. D. (1973), [*Strong rigidity of locally symmetric spaces*](https://books.google.com/books?id=xT0SFmrFrWoC), Annals of mathematics studies, vol. 78, [Princeton University Press](/source/Princeton_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-691-08136-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08136-6), [MR](/source/MR_(identifier)) [0385004](https://mathscinet.ams.org/mathscinet-getitem?mr=0385004)

- Prasad, Gopal (1973), "Strong rigidity of Q-rank 1 lattices", *[Inventiones Mathematicae](/source/Inventiones_Mathematicae)*, **21** (4): 255–286, [Bibcode](/source/Bibcode_(identifier)):[1973InMat..21..255P](https://ui.adsabs.harvard.edu/abs/1973InMat..21..255P), [doi](/source/Doi_(identifier)):[10.1007/BF01418789](https://doi.org/10.1007%2FBF01418789), [ISSN](/source/ISSN_(identifier)) [0020-9910](https://search.worldcat.org/issn/0020-9910), [MR](/source/MR_(identifier)) [0385005](https://mathscinet.ams.org/mathscinet-getitem?mr=0385005), [S2CID](/source/S2CID_(identifier)) [55739204](https://api.semanticscholar.org/CorpusID:55739204)

- [Spatzier, R. J.](/source/Ralf_J._Spatzier) (1995), "Harmonic Analysis in Rigidity Theory", in Petersen, Karl E.; Salama, Ibrahim A. (eds.), *Ergodic Theory and its Connection with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference*, Cambridge University Press, pp. 153–205, [ISBN](/source/ISBN_(identifier)) [0-521-45999-0](https://en.wikipedia.org/wiki/Special:BookSources/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.)*

- [Thurston, William](/source/William_Thurston) (1978–1981), [*The geometry and topology of 3-manifolds*](https://library.slmath.org/nonmsri/gt3m/PDF/13.pdf) (PDF), Princeton lecture notes. (Gives two proofs: one similar to Mostow's original proof, and another based on the [Gromov norm](/source/Gromov_norm))

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Adapted from the Wikipedia article [Mostow rigidity theorem](https://en.wikipedia.org/wiki/Mostow_rigidity_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mostow_rigidity_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.