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While the theorem shows that there is no deformation space of (complete) hyperbolic structures on a finite volume hyperbolic ''n''-manifold (for n >2), 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 was proven for the closed case by G. D. Mostow in 1968 and extended to the finite volume case by G. Prasad (and independently Marden). Sometimes the theorem is called the Mostow-Prasad rigidity theorem. This theorem has been generalized to non-uniform lattices. An important alternate proof using the Gromov Norm was given by M. Gromov in 1979 . 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 Here, is the Fundamental Group of the manifold. 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 Discrete subgroups of the isometry group of hyperbolic APPLICATIONS An important corollary is that a finite volume hyperbolic ''n''-manifold ''M'' for n > 2 has no nontrivial inner automorphisms of . One can conclude that the group of isometries of ''M'' is finite and isomorphic to . REFERENCES
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