{{Short description|Theorem in hyperbolic geometry}} 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 manifold]]s by {{harvs|txt|authorlink=George Mostow|last=Mostow|year= 1968}} and extended to finite volume manifolds by {{harvtxt|Marden|1974}} in 3 dimensions, and by {{harvs|txt|authorlink=Gopal Prasad|last=Prasad|year=1973}} in all dimensions at least 3. {{harvtxt|Gromov|1981}} gave an alternate proof using the [[Gromov norm]]. {{harvtxt|Besson|Courtois|Gallot|url=https://www.researchgate.net/profile/Gilles_Courtois2/publication/231902765_Minimal_entropy_and_Mostow's_rigidity_theorems/links/02e7e538a32469eacc000000.pdf|1996}} gave the simplest available proof.
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic <math>n</math>-manifold (for <math>n >2</math>) is a point, for a hyperbolic surface of [[genus (mathematics)|genus]] <math>g>1</math> there is a [[moduli space]] of dimension <math>6g-6</math> that parameterizes all metrics of constant curvature (up to [[diffeomorphism]]), a fact essential for [[Teichmüller 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 group]]s).
===Geometric form===
Let <math>\mathbb H^n</math> be the <math>n</math>-dimensional [[hyperbolic space]]. A complete hyperbolic manifold can be defined as a quotient of <math>\mathbb H^n</math> by a group of isometries acting freely and [[Group action (mathematics)#Types of action|properly discontinuously]] (it is equivalent to define it as a [[hyperbolic manifold|Riemannian manifold with sectional curvature -1]] which is [[Riemannian manifold#Riemannian manifolds as metric spaces|complete]]). It is of finite volume if the integral of a [[volume form]] is finite (which is the case, for example, if it is compact). The Mostow rigidity theorem may be stated as:
:''Suppose <math>M</math> and <math>N</math> are complete finite-volume hyperbolic manifolds of dimension <math>n \ge 3</math>. If there exists an [[isomorphism]] <math>f\colon \pi_1(M) \to \pi_1(N)</math> then it is induced by a unique isometry from <math>M</math> to <math>N</math>.''
Here <math>\pi_1(X)</math> is the [[fundamental group]] of a manifold <math>X</math>. If <math>X</math> is a hyperbolic manifold obtained as the quotient of <math>\mathbb H^n</math> by a group <math>\Gamma</math> then <math>\pi_1(X) \cong \Gamma</math>.
An equivalent statement is that any [[homotopy equivalence]] from <math>M</math> to <math>N</math> can be homotoped to a unique isometry. The proof actually shows that if <math>N</math> has greater dimension than <math>M</math> then there can be no homotopy equivalence between them.
===Algebraic form=== The group of isometries of hyperbolic space <math>\mathbb H^n</math> can be identified with the Lie group <math>\mathrm{PO}(n,1)</math> (the [[projective orthogonal group]] of a [[Quadratic form#Real quadratic forms|quadratic form of signature]] <math>(n,1)</math>. Then the following statement is equivalent to the one above.
:''Let <math> n \ge 3 </math> and <math>\Gamma</math> and <math>\Lambda</math> be two [[Lattice (discrete subgroup)|lattices]] in <math>\mathrm{PO}(n,1)</math> and suppose that there is a group isomorphism <math>f\colon \Gamma \to \Lambda</math>. Then <math>\Gamma</math> and <math>\Lambda</math> are conjugate in <math>\mathrm{PO}(n,1)</math>. That is, there exists a <math>g \in \mathrm{PO}(n,1)</math> such that <math> \Lambda = g \Gamma g^{-1}</math>. ''
=== 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 space]]s of dimension at least three, or in its algebraic formulation for all lattices in [[simple Lie group]]s not locally isomorphic to <math>\mathrm{SL}_2(\R)</math>.
==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 <math>\operatorname{Out}(\pi_1(M))</math>.
Mostow rigidity was also used by Thurston to prove the uniqueness of [[circle packing theorem|circle packing representations]] of [[planar graph|triangulated planar graphs]].{{sfn|Thurston|1978–1981|loc=Chapter 13}}
A consequence of Mostow rigidity of interest in [[geometric group theory]] is that there exist [[hyperbolic group]]s which are [[Quasi-isometry|quasi-isometric]] but not [[Commensurability (group theory)|commensurable]] to each other.
== See also ==
* [[Superrigidity]], a stronger result for higher-rank spaces * [[Local rigidity]], a result about deformations that are not necessarily lattices.
==Notes== {{reflist}}
==References==
*{{Citation |last1=Besson |first1=Gérard|last2=Courtois|first2=Gilles|last3=Gallot|first3=Sylvestre|author-link3=Sylvestre Gallot|title=Minimal entropy and Mostow's rigidity theorems|journal=Ergodic Theory and Dynamical Systems|volume=16|issue=4|year=1996|pages=623–649|doi=10.1017/S0143385700009019|s2cid=122773907 }} *{{Citation | last1=Gromov | first1=Michael | title=Bourbaki Seminar, Vol. 1979/80 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-10292-2 | doi=10.1007/BFb0089927 | mr=636516 | year=1981 | volume=842 | chapter=Hyperbolic manifolds (according to Thurston and Jørgensen) | chapter-url=http://www.numdam.org/numdam-bin/fitem?id=SB_1979-1980__22__40_0 | pages=40–53 | url-status=dead | archive-url=https://web.archive.org/web/20160110061753/http://www.numdam.org/numdam-bin/fitem?id=SB_1979-1980__22__40_0 | archive-date=2016-01-10 | url=https://www.numdam.org/article/SB_1979-1980__22__40_0.pdf }} *{{Citation | last1=Marden | first1=Albert | title=The geometry of finitely generated kleinian groups | jstor=1971059 | mr=0349992 | zbl = 0282.30014 | year=1974 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=99 | issue=3 | pages=383–462 | doi=10.2307/1971059}} *{{Citation|first=G. D.|last=Mostow|url=https://www.numdam.org/item?id=PMIHES_1968__34__53_0|title=Quasi-conformal mappings in ''n''-space and the rigidity of the hyperbolic space forms|journal= Publ. Math. IHÉS|volume=34|year=1968|pages=53–104|doi=10.1007/bf02684590|s2cid=55916797 }} *{{Citation | last1=Mostow | first1=G. D. | title=Strong rigidity of locally symmetric spaces | url=https://books.google.com/books?id=xT0SFmrFrWoC | publisher=[[Princeton University Press]] | series=Annals of mathematics studies | isbn=978-0-691-08136-6 | mr=0385004 | year=1973 | volume=78}} *{{Citation | last1=Prasad | first1=Gopal | title=Strong rigidity of Q-rank 1 lattices | doi=10.1007/BF01418789 | mr=0385005 | year=1973 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=21 | issue=4 | pages=255–286| bibcode=1973InMat..21..255P | s2cid=55739204 }} *{{citation|first=R. J.|last=Spatzier|author-link=Ralf J. Spatzier|contribution=Harmonic Analysis in Rigidity Theory|pages=153–205|title=Ergodic Theory and its Connection with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference|editor1-first=Karl E.|editor1-last=Petersen|editor2-first=Ibrahim A.|editor2-last=Salama|publisher=Cambridge University Press|year=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.)'' *{{citation|first=William|last=Thurston|author-link=William Thurston|url=https://library.slmath.org/nonmsri/gt3m/PDF/13.pdf|title=The geometry and topology of 3-manifolds|publisher=Princeton lecture notes|year=1978–1981}}. (Gives two proofs: one similar to Mostow's original proof, and another based on the [[Gromov norm]])
[[Category:Hyperbolic manifolds]] [[Category:Theorems in differential geometry]]