Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3

Joël Dubois

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 2, page 535-551
  • ISSN: 0373-0956

Abstract

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We introduce a class of knots and use it to prove a topological rigidity criterion for homotopy equivalences between 3-manifolds. As an application, we give a new proof of Gabai’s virtual rigidity theorem for hyperbolic 3-manifolds.

How to cite

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Dubois, Joël. "Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3." Annales de l'institut Fourier 48.2 (1998): 535-551. <http://eudml.org/doc/75292>.

@article{Dubois1998,
abstract = {En définissant une nouvelle classe de nœuds dans les variétés de dimension 3, on obtient une démonstration plus classique du théorème de rigidité virtuelle des variétés hyperboliques de D. Gabai.},
author = {Dubois, Joël},
journal = {Annales de l'institut Fourier},
keywords = {Fox group; Fox-residually nilpotent; Gabai's theorem},
language = {fre},
number = {2},
pages = {535-551},
publisher = {Association des Annales de l'Institut Fourier},
title = {Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3},
url = {http://eudml.org/doc/75292},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Dubois, Joël
TI - Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 535
EP - 551
AB - En définissant une nouvelle classe de nœuds dans les variétés de dimension 3, on obtient une démonstration plus classique du théorème de rigidité virtuelle des variétés hyperboliques de D. Gabai.
LA - fre
KW - Fox group; Fox-residually nilpotent; Gabai's theorem
UR - http://eudml.org/doc/75292
ER -

References

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  1. [Du] J. DUBOIS, Nœuds géométriquement-libres et rigidité topologique des variétés de dimension 3, Thèse, Université Toulouse-III, 1996. 
  2. [Ep] D.B.A. EPSTEIN, The degree of map, Proc. London Math. Soc., (3) 16 (1966), 369-383. Zbl0148.43103MR33 #700
  3. [Ga1] D. GABAI, Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc., 7 (1994), 193-198. Zbl0801.57009MR94b:57016
  4. [Ga2] D. GABAI, Foliations and the topology of 3-manifolds, J. Differential Geom., 18 (1983), 445-503. Zbl0533.57013MR86a:57009
  5. [GMT] D. GABAI, G.R. MEYERHOFF, N. THURSTON, Homotopy Hyperbolic 3-manifolds are hyperbolic, preprint. Zbl1052.57019
  6. [He] J. HEMPEL, 3-Manifolds, Ann. of Math. Stud., vol 86, Princeton Univ. Press, Princeton NJ, 1976. Zbl0345.57001MR54 #3702
  7. [Hi] M.W. HIRSCH, Differential topology, Springer-Verlag, Berlin-Heidelberg-New York, 1976. Zbl0356.57001MR56 #6669
  8. [HS] J. HASS, P. SCOTT, Homotopy equivalence and homeomorphism of 3-manifolds, Topology, 31 (1992), 493-517. Zbl0771.57007MR94g:57021
  9. [Ja] W. JACO, Lecture on 3-manifold topology, CBMS Lecture Notes, No. 43, Amer. Math. Soc., Providence, RI, 1980. Zbl0433.57001MR81k:57009
  10. [Jo] K. JOHANNSON, Homotopy equivalences of 3-manifolds with boundaries, Lecture Note in Math. Vol. 761 (Springer, Berlin-Heidelberg-New York, 1979). Zbl0412.57007MR82c:57005
  11. [JS] W. JACO, P. SHALEM, Seifert fibred spaces in 3-manifolds, Mem. Amer. Math. Soc., 220 (1980). Zbl0415.57005
  12. [Kr] P.H. KROPHOLLER, A note on centrality in 3-manifold groups, School of Math. Sci., Queen Mary College, London E1 4NS (1989), 261-266. 
  13. [Ma] W. MAGNUS, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann., 111 (1935). Zbl0011.15201JFM61.0102.02
  14. [Ma1] A. MAL'CEV, On isomorphic matrix representations of infinite groups, Math. Sb., 8 (50) (1940), 405-422. Zbl0025.00804MR2,216dJFM66.0088.03
  15. [Sa] T. SAKAI, Geodesic knots in hyperbolic 3-manifold, Kobe J. Math., 8 (1991), 81-87. Zbl0749.57003MR93a:57011
  16. [Sc1] P. SCOTT, The geometry of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. Zbl0561.57001MR84m:57009
  17. [Sc2] P. SCOTT, There are no fake Seifert fibred spaces with infinite π1, Ann. of Math., (2) 117 (1983), 35-70. Zbl0516.57006MR84c:57008
  18. [St] J. STALLINGS, Homology and Central Series of Groups, J. Algebra, 2 (1965), 170-181. Zbl0135.05201MR31 #232
  19. [Th1] W. THURSTON, Geometry and topology of 3-manifolds, Lecture Notes, Princeton University, Princeton NJ, 1978-1979. 
  20. [Th2] W. THURSTON, Three dimensionnal manifolds, Kleinian groups, and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982) 357-381. Zbl0496.57005
  21. [Wa1] F. WALDHAUSEN, On irreducible 3-manifolds which are sufficiently large, Ann. of Math., (2) 87 (1968), 56-88. Zbl0157.30603MR36 #7146
  22. [Wa2] F. WALDHAUSEN, On the determination of some bounded 3-manifold by their fundamental groups alone, Proc. of Int. Sym. of Topology, Hercy-Novi, Yugoslavia, 1968 : Beograd (1969), 331-332. Zbl0202.54702
  23. [Wr] A.H. WRIGHT, Monotone mappings and degree one mappings between pl manifolds, Geometry Topology (Proc. Conf., Park City, Utah, 1974), Lecture Note in Math. Vol 438, Springer, Berlin (1975) 441-459. Zbl0309.55006MR52 #15489

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