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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    BOILEAU, MICHEL NI, YI and WANG, SHICHENG 2008. ON STANDARD FORMS OF 1-DOMINATIONS BETWEEN KNOTS WITH SAME GROMOV VOLUMES. Communications in Contemporary Mathematics, Vol. 10, Issue. supp01, p. 857.


    Kwasik, Sławomir and Schultz, Reinhard 2007. Cartesian powers of 3-manifolds. Topology and its Applications, Vol. 154, Issue. 1, p. 176.


    Derbez, Pierre 2003. A criterion for homeomorphism between closed Haken manifolds. Algebraic & Geometric Topology, Vol. 3, Issue. 1, p. 335.


    Rong, Yongwu 1995. Degree one maps of Seifert manifolds and a note on Seifert volume. Topology and its Applications, Vol. 64, Issue. 2, p. 191.


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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 118, Issue 1
  • July 1995, pp. 141-160

A rigidity theorem for Haken manifolds

  • Teruhiko Soma (a1)
  • DOI: http://dx.doi.org/10.1017/S0305004100073527
  • Published online: 24 October 2008
Abstract

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
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