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A rigidity theorem for Haken manifolds

  • Teruhiko Soma (a1)

A compact, orientable 3-manifold M is called hyperbolic if int M admits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifold M is irreducible, and each component of ∂M is an incompressible torus. Let f: MN be a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], if f is π1-isomorphic then f is properly homotopic to a diffeomorphism g: MN such that g | int M: int M → int N is isometric. In particular, the topological type of int M determines uniquely the hyperbolic structure on int M up to isometry, so the volume vol (int M) of int M is well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous map f:MN between hyperbolic 3-manifolds with vol(int M) = deg(f) vol(int N) is properly homotopic to a deg(f)-fold covering g:MN such that g | int M: int M → int N is locally isometric.

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[1] L. V. Ahlfors . Finitely generated Kleinian groups. Amer. J. Math. 86 (1964), 229236.

[2] F. Bonahon . Bouts des variétés hyperboliques de dimension 3. Ann. of Math. 124 (1986), 71158.

[12] W. Thurston . Three dimensional manifolds, Kleinian groups, and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357381.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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