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A rigidity theorem for Haken manifolds

Published online by Cambridge University Press:  24 October 2008

Teruhiko Soma
Teruhiko Soma
Affiliation:
Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Saitama-ken 350–03, Japan
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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: M → N 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: M → N 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:M → N between hyperbolic 3-manifolds with vol(int M) = deg(f) vol(int N) is properly homotopic to a deg(f)-fold covering g:M → N such that g | int M: int M → int N is locally isometric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 141 - 160
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Ahlfors, L. V.. Finitely generated Kleinian groups. Amer. J. Math. 86 (1964), 229–236.CrossRefGoogle Scholar
[2]Bonahon, F.. Bouts des variétés hyperboliques de dimension 3. Ann. of Math. 124 (1986), 71–158.CrossRefGoogle Scholar
[3]Gromov, M.. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1983), 213–307.Google Scholar
[4]Hempel, J.. 3-manifolds. Ann. of Math. Studies 86 (Princeton Univ. Press, 1976).Google Scholar
[5]Jaco, W.. Lectures on three-manifold topology. C.B.M.S. Regional Conf. Ser. in Math. no. 43 (Amer. Math. Soc., 1980).CrossRefGoogle Scholar
[6]Jaco, W. and Shalen, P.. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. no. 220 (1979).CrossRefGoogle Scholar
[7]Johannson, K.. Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics Vol. 761 (Springer, 1979).CrossRefGoogle Scholar
[8]Mostow, G. D.. Strong rigidity of locally symmetric spaces. Ann. of Math. Studies 78 (Princeton Univ. Press, 1973).Google Scholar
[9]Rong, Y.. Degree one maps between geometric 3-manifolds. Trans. Amer. Math. Soc. 322 (1992), 411–436.CrossRefGoogle Scholar
[10]Soma, T.. Virtual fibre groups in 3-manifold groups. J. London Math. So. (2) 43 (1991), 337–354.CrossRefGoogle Scholar
[11]Thurston, W.. The geometry and topology of 3-manifolds. Mimeographed Notes (Princeton Univ., 1977/1978).Google Scholar
[12]Thurston, W.. Three dimensional manifolds, Kleinian groups, and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357–381.CrossRefGoogle Scholar