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On the geometric and topological rigidity of hyperbolic 3-manifolds

Author: David Gabai
Journal: Bull. Amer. Math. Soc. 31 (1994), 228-232
MSC: Primary 57M50; Secondary 57N10
MathSciNet review: 1261238
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Abstract: A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition.

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  • [Bo] F. Bonahon, Diffeotopes des espaces lenticulaires, Topology 22 (1983), 305-314. MR 710104 (85d:57008)
  • [BS] F. Bonahon and L. Siebenmann (to appear).
  • [FJ] F. T. Farrell and L. Jones, A topological analogue of Mostow's rigidity theorem, J. Amer. Math. Soc. 2 (1989), 257-370. MR 973309 (90h:57023a)
  • [Fr] W. Franz, Abbildungsklassen und fixpunktklassen dreidimensionalen linsenraume, J. Reine. Angew. Math. 185 (1943), 65-77. MR 0009108 (5:103e)
  • [G1] Gabai, Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc. 7 (1994), 193-198. MR 1205445 (94b:57016)
  • [G2] -, On the geometric and topological rigidity of hyperbolic 3-manifolds, preprint.
  • [GM1] F. Gehring and G. Martin, Commutators, collars and the geometry of Mobius groups, J. d'Analyse (to appear). MR 1269219 (96c:30040)
  • [GM2] -, Torsion and volume in hyperbolic 3-folds, in preparation.
  • [Gr] M. Gromov, Hyperbolic manifolds according to Thurston and Jorgensen, Sem. Bourbaki 32 (1979), 40-52. MR 636516 (84b:53046)
  • [Me] R. Meyerhoff, A lower bound for the volume of hyperbolic 3-manifolds, Canada. J. Math. 39 (1987), 1038-1056. MR 918586 (88k:57049)
  • [Mo] G. D. Mostow, Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 0236383 (38:4679)
  • [Ol] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458-480. MR 0058212 (15:338a)
  • [Re] K. Reidemeister, Homotopieringe und Linsenraume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109.
  • [Ru] M. RuefF, Beitrage zur untersuchung der abbildungen von mannigfaltigkeiten, Compositio Math. 6 (1938), 161-202. MR 1557021
  • [S] P. Scott, There are no fake Seifert fibred spaces with infinite $ {\pi _1}$, Ann. of Math. (2) 117 (1983), 35-70. MR 683801 (84c:57008)
  • [T] W. P. Thurston, Three-dimensional manifolds, Kleinian groups, and hyperbolic geometry, Proc. Sympos. Pure Math., vol. 39, Amer. Math. Soc., Providence, RI, 1983, pp. 87-111. MR 648524 (83h:57019)
  • [W] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 0224099 (36:7146)
  • [We] J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3-manifolds, available by anonymous ftp from

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