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 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 {{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 $n$ -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. 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).

=Geometric form=

Let $\mathbb H^n$ be the $n$ -dimensional hyperbolic space. A complete hyperbolic manifold can be defined as a quotient of $\mathbb H^n$ by a group of isometries acting freely and properly discontinuously (it is equivalent to define it as a Riemannian manifold with sectional curvature -1 which is 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 $M$ and $N$ are complete finite-volume hyperbolic manifolds of dimension $n \ge 3$ . If there exists an isomorphism $f\colon \pi_1(M) \to \pi_1(N)$ then it is induced by a unique isometry from $M$ to $N$ .

Here $\pi_1(X)$ is the fundamental group of a manifold $X$ . If $X$ is an hyperbolic manifold obtained as the quotient of $\mathbb H^n$ by a group $\Gamma$ then $\pi_1(X) \cong \Gamma$ .

An equivalent statement is 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=

The group of isometries of hyperbolic space $\mathbb H^n$ can be identified with the Lie group $\mathrm{PO}(n,1)$ (the projective orthogonal group of a quadratic form of signature $(n,1)$ . Then the following statement is equivalent to the one above.

:Let $n \ge 3$ and $\Gamma$ and $\Lambda$ be two lattices in $\mathrm{PO}(n,1)$ and suppose that there is a group isomorphism $f\colon \Gamma \to \Lambda$ . Then $\Gamma$ and $\Lambda$ are conjugate in $\mathrm{PO}(n,1)$ . That is, there exists a $g \in \mathrm{PO}(n,1)$ such that $\Lambda = g \Gamma g^{-1}$ .

= 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 of dimension at least three, or in its algebraic formulation for all lattices in simple Lie groups not locally isomorphic to $\mathrm{SL}_2(\R)$ .

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 $\operatorname{Out}(\pi_1(M))$ .

Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations of 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 groups which are quasi-isometric but not commensurable to each other.

Notes

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=http://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=http://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=http://www.msri.org/publications/books/gt3m/|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 geometry

Category:Theorems in differential geometry