Mostow rigidity theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Mostow_rigidity_theorem


00:01:35 1 The theorem
00:02:00 1.1 Geometric form
00:06:01 1.2 Algebraic form
00:09:03 1.3 In greater generality
00:09:50 2 Applications
00:10:42 3 See also



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SUMMARY
=======
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.
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic 
  
    
      
        n
      
    
    {\displaystyle 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.