Mostow rigidity theorem
Encyclopedia
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 and extended to finite volume manifolds by in 3-dimensions, and by in dimensions at least 3. gave an alternate proof using the Gromov norm
.
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.
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.
Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations
of triangulated planar graphs
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
of dimension greater than two is determined by the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
and hence unique. The theorem was proven for closed manifolds by and extended to finite volume manifolds by in 3-dimensions, and by in dimensions at least 3. gave an alternate proof using the Gromov norm
Gromov norm
In mathematics, the Gromov norm of a compact oriented n-manifold is a norm on the homology given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle...
.
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
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
g > 1 there is a moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of dimension 6g − 6 that parameterizes all metrics of constant curvature (up to diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
), a fact essential for Teichmüller theory. In dimension three, there is a "non-rigidity" theorem due to Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
called the hyperbolic Dehn surgery
Hyperbolic Dehn surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold...
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-manifoldsHyperbolic manifoldIn mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
with n > 2. If there exists an isomorphismIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
ƒ : π1(M) → π1(N) then it is induced by a unique isometry from M to N.
Here, π1(M) is the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
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 discreteDiscrete groupIn mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
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
Circle packing theorem
The circle packing theorem describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles whose interiors are disjoint...
of triangulated planar graphs
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...
.