Mostow rigidity theorem

In mathematics, the Mostowsche rigidity theorem states (also strong rigidity theorem, or Mostow - Prasad rigidity theorem ), in essence, that the geometry of a hyperbolic manifold finite volume of the dimension is greater than 2 determined by its fundamental group and is therefore unique. The theorem was proved for closed manifolds of George Mostow, then extended to finite volume manifolds of dimension 3 in Albert Marden and Gopal Prasad in dimension. Gromov gave another proof using the simplicial volume. In André Weil a weaker local version goes back, namely that kokompakte discrete groups of isometries of hyperbolic space of dimension at least 3 do not allow non-trivial deformations. A worsening of the Mostowschen rigidity theorem is proved by Margulis super- rigidity theorem.

The theorem states that the deformation space of (complete) hyperbolic structures on a hyperbolic n -manifold finite volume ( ) is a point. In contrast, a hyperbolic surface of genus g has a 6g - 6-dimensional moduli space, which classifies the metrics of constant curvature (up to diffeomorphism ), see Teichmüller space. In dimension 3, there is a " flexibility set" of Thurston, the theorem on hyperbolic Dehn surgery: it allows finite volume hyperbolic structures to deform if you allow changes to the topology of the manifold. There is also an extensive theory of deformations of hyperbolic structures on hyperbolic manifolds of infinite volume.

Rigidity theorem

The theorem can be formulated in a geometric or algebraic version.

Geometric formulation

Herein, π1 (M) the fundamental group of the manifold M.

An equivalent version states that every homotopy equivalence is homotopic to a unique isometry.

Algebraic formulation

An equivalent version is:

Generalization: Thurston's rigidity theorem

When m and n are whole hyperbolic manifolds finite volume of the dimension > 2 and is an integer d, the relationship

Holds, then every map from the mapping degree d homotopic to a locally isometric - d -fold storage.

In particular, it follows from Vol ( M) = v (N) that any map from the mapping degree 1 is homotopic to an isometry.

Applications

The group of isometries of a hyperbolic n -manifold finite volume M (for n> 2) is finite and isomorphic to Out ( π1 (M)).

Thurston used Mostow rigidity - to show the uniqueness of triangulated planar graphs associated to circle packings.