# Mostow rigidity theorem

Last updated

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 Mostow  ( 1968 ) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad  ( 1973 ) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois & Gallot (1996) gave the simplest available proof.

## Contents

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 $n\geq 3$ . 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 $n\geq 3$ 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 $\mathrm {SL} _{2}(\mathbb {R} )$ .

## 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 $\operatorname {Out} (\pi _{1}(M))$ .

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

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.

• Superrigidity, a stronger result for higher-rank spaces
• Local rigidity, a result about deformations that are not necessarily lattices.

## Related Research Articles In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory. In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries. In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

In mathematics, the Teichmüller space of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

In differential geometry, the Margulis lemma is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold. Roughly, it states that within a fixed radius, usually called the Margulis constant, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group. In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.

In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.

In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.

In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a -structure are always manifolds and are called (G, X)-manifolds. This notion is often used with G being a Lie group and X a homogeneous space for G. Foundational examples are hyperbolic manifolds and affine manifolds.

Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker than superrigidity.

Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group . They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.

In Lie theory, an area of mathematics, the Kazhdan–Margulis theorem is a statement asserting that a discrete subgroup in semisimple Lie groups cannot be too dense in the group. More precisely, in any such Lie group there is a uniform neighbourhood of the identity element such that every lattice in the group has a conjugate whose intersection with this neighbourhood contains only the identity. This result was proven in the sixties by David Kazhdan and Grigory Margulis.

In the mathematical subject of geometric group theory, the Švarc–Milnor lemma is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space , is quasi-isometric to .

In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.

In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces.

• Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre (1996), "Minimal entropy and Mostow's rigidity theorems", Ergodic Theory and Dynamical Systems, 16 (4): 623–649, doi:10.1017/S0143385700009019
• Gromov, Michael (1981), "Hyperbolic manifolds (according to Thurston and Jørgensen)", Bourbaki Seminar, Vol. 1979/80 (PDF), Lecture Notes in Math., vol. 842, Berlin, New York: Springer-Verlag, pp. 40–53, doi:10.1007/BFb0089927, ISBN   978-3-540-10292-2, MR   0636516, archived from the original on 2016-01-10
• Marden, Albert (1974), "The geometry of finitely generated kleinian groups", Annals of Mathematics , Second Series, 99 (3): 383–462, doi:10.2307/1971059, ISSN   0003-486X, JSTOR   1971059, MR   0349992, Zbl   0282.30014
• Mostow, G. D. (1968), "Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms", Publ. Math. IHES, 34: 53–104, doi:10.1007/bf02684590
• Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals of mathematics studies, vol. 78, Princeton University Press, ISBN   978-0-691-08136-6, MR   0385004
• Prasad, Gopal (1973), "Strong rigidity of Q-rank 1 lattices", Inventiones Mathematicae , 21 (4): 255–286, doi:10.1007/BF01418789, ISSN   0020-9910, MR   0385005
• Spatzier, R. J. (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   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 (1978–1981), The geometry and topology of 3-manifolds, Princeton lecture notes. (Gives two proofs: one similar to Mostow's original proof, and another based on the Gromov norm)