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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

On the geometric and topological rigidity of hyperbolic 3-manifolds

Author(s): David Gabai
Journal: Bull. Amer. Math. Soc. 31 (1994), 228-232.
MathSciNet review: 1261238
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Abstract | References | Additional information

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.


References:

Bibliography

[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 geom.umn.edu.


Additional Information:

DOI: 10.1090/S0273-0979-1994-00523-3
PII: S 0273-0979(1994)00523-3
Copyright of article: Copyright 1994, American Mathematical Society




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