Mostow rigidity

In mathematics , Mostow 's law of rigidity (also known as the strong law of rigidity or Mostow-Prasad's law of rigidity ) essentially states that the geometry of a hyperbolic manifold of finite volume of dimension greater than 2 is determined by its fundamental group and is therefore unique. The theorem was proved for closed manifolds by George Mostow , then extended to manifolds of finite volume by Albert Marden in dimension 3 and by Gopal Prasad in dimension . Gromov gave another proof with the help of simplicial volume . A weaker local version goes back to André Weil , namely that cocompact discrete groups of isometries of the hyperbolic space of dimension at least 3 do not allow any non-trivial deformations. A tightening of Mostow's law of rigidity is the super- rigidity theorem proved by Margulis . ${\ displaystyle \ geq 3}$

The theorem says that the deformation space of the (complete) hyperbolic structures on a hyperbolic n-manifold of finite volume ( ) is a point. In contrast, a hyperbolic surface of gender g has a 6g-6-dimensional module space that classifies the metrics of constant curvature (except for diffeomorphism), see Teichmüller space . In dimension 3 there is a "flexibility theorem " by Thurston , the theorem about hyperbolic stretching surgery : it allows hyperbolic structures of finite volume to be deformed if one allows changes in the topology of the manifold. There is also an extensive theory of the deformations of hyperbolic structures on hyperbolic manifolds of infinite volume. ${\ displaystyle n> 2}$ ${\ displaystyle K = -1}$

Rigidity Theorem

The sentence can be formulated in geometric or algebraic form.

Geometric formulation

Let and be complete hyperbolic n-manifolds of finite volume with . If there is an isomorphism , then it is induced by a unique isometry ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle n> 2}$ ${\ displaystyle \ pi _ {1} M \ rightarrow \ pi _ {1} N}$ ${\ displaystyle M \ rightarrow N}$ .

Here denotes the fundamental group of the manifold . ${\ displaystyle \ pi _ {1} M}$ ${\ displaystyle M}$

An equivalent version says that every homotopy equivalence is homotopic to a unique isometry. ${\ displaystyle M \ rightarrow N}$

Algebraic formulation

An equivalent version is:

Let and be discrete subgroups of the isometric group of hyperbolic -space with , whose quotients and have finite volume. If and as groups are isomorphic , then they are conjugate subsets of the isometric group . ${\ displaystyle \ Gamma}$ ${\ displaystyle \ Delta}$ ${\ displaystyle n}$ ${\ displaystyle H ^ {n}}$ ${\ displaystyle n> 2}$ ${\ displaystyle H ^ {n} \ backslash \ Gamma}$ ${\ displaystyle H ^ {n} \ backslash \ Delta}$ ${\ displaystyle \ Gamma}$ ${\ displaystyle \ Delta}$ ${\ displaystyle isom (H ^ {n})}$

Generalization: Thurston's Theorem of Rigidity

If and are complete hyperbolic manifolds of finite volume of the dimension and for an integer the relation ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle> 2}$ ${\ displaystyle d}$

V

{\ displaystyle ol (M) = dVol (N)}

holds, then every mapping of the degree of mapping is homotopic to a local-isometric -fold overlay . ${\ displaystyle d}$ ${\ displaystyle d}$

In particular, it follows from this that every mapping with mapping degree 1 is homotopic to an isometry. ${\ displaystyle Vol (M) = Vol (N)}$

Applications

The group of isometries of a hyperbolic -manifold of finite volume is always finite and isomorphic to . ${\ displaystyle n}$ ${\ displaystyle M}$ ${\ displaystyle n> 2}$ ${\ displaystyle Out (\ pi _ {1} M)}$

Thurston used Mostow's rigidity to show the uniqueness of the circular packing associated with triangulated planar graphs .

literature

Gromov, Michael: Hyperbolic manifolds (according to Thurston and Jørgensen). (PDF; 1.1 MB) Bourbaki Seminar, Vol. 1979/80, pp. 40–53, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981.
Mostow, GD: Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, No. 78. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1973.
Spatzier, RJ: Harmonic analysis in rigidity theory. (PDF; 412 kB) Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 153-205, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.
William Thurston, The geometry and topology of 3-manifolds , Princeton lecture notes (1978-1981). (Represents both proofs: one similar to Mostow's original proof, another using Gromow's simplicial volume.)

Web links

Lücker: Approaches to Mostow rigidity in hyperbolic space
Bourdon: Mostow type rigidity theorems

Individual evidence

^ GD Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms , Publ. Math. IHES 34 (1968) 53-104
^ Marden, Albert: The geometry of finitely generated kleinian groups. Ann. of Math. (2) 99: 383-462 (1974).
^ Prasad, Gopal: Strong rigidity of Q-rank 1 lattices. (PDF; 1.4 MB) Invent. Math. 21: 255-286 (1973).
↑ Gromov, Michail: Volume and bounded cohomology. (PDF; 9.9 MB) Inst. Hautes Études Sci. Publ. Math. No. 1982, 56: 5-99
↑ Weil, André: On discrete subgroups of Lie groups. I: Ann. of Math. (2) 72 1960 369-384 pdf ; II: Ann. of Math. (2) 75 1962 578-602 pdf .

[1] GD Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms , Publ. Math. IHES 34 (1968) 53-104

[2] Marden, Albert: The geometry of finitely generated kleinian groups. Ann. of Math. (2) 99: 383-462 (1974).

[3] Prasad, Gopal: Strong rigidity of Q-rank 1 lattices. (PDF; 1.4 MB) Invent. Math. 21: 255-286 (1973).

[4] Gromov, Michail: Volume and bounded cohomology. (PDF; 9.9 MB) Inst. Hautes Études Sci. Publ. Math. No. 1982, 56: 5-99

[5] Weil, André: On discrete subgroups of Lie groups. I: Ann. of Math. (2) 72 1960 369-384 pdf ; II: Ann. of Math. (2) 75 1962 578-602 pdf .