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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 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 6''g'' − 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.
==The theorem==
The theorem can be given in a geometric formulation, and in an algebraic formulation.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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