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

Table of contents for this issue | Previous article | Next article

Joël Dubois
Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3
Annales de l'institut Fourier, 48 no. 2 (1998), p. 535-551, doi: 10.5802/aif.1628
Article PDF | Reviews MR 2000a:57027 | Zbl 0899.57008

Résumé - Abstract

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.

Bibliography

[Du] J. DUBOIS, Nuds géométriquement-libres et rigidité topologique des variétés de dimension 3, Thèse, Université Toulouse-III, 1996.
[Ep] D.B.A. EPSTEIN, The degree of map, Proc. London Math. Soc., (3) 16 (1966), 369-383. MR 33 #700 | Zbl 0148.43103
[Ga1] D. GABAI, Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc., 7 (1994), 193-198. MR 94b:57016 | Zbl 0801.57009
[Ga2] D. GABAI, Foliations and the topology of 3-manifolds, J. Differential Geom., 18 (1983), 445-503. MR 86a:57009 | Zbl 0533.57013
[GMT] D. GABAI, G.R. MEYERHOFF, N. THURSTON, Homotopy Hyperbolic 3-manifolds are hyperbolic, preprint.
[He] J. HEMPEL, 3-Manifolds, Ann. of Math. Stud., vol 86, Princeton Univ. Press, Princeton NJ, 1976. MR 54 #3702 | Zbl 0345.57001
[Hi] M.W. HIRSCH, Differential topology, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 56 #6669 | Zbl 0356.57001
[HS] J. HASS, P. SCOTT, Homotopy equivalence and homeomorphism of 3-manifolds, Topology, 31 (1992), 493-517. MR 94g:57021 | Zbl 0771.57007
[Ja] W. JACO, Lecture on 3-manifold topology, CBMS Lecture Notes, No. 43, Amer. Math. Soc., Providence, RI, 1980. MR 81k:57009 | Zbl 0433.57001
[Jo] K. JOHANNSON, Homotopy equivalences of 3-manifolds with boundaries, Lecture Note in Math. Vol. 761 (Springer, Berlin-Heidelberg-New York, 1979). MR 82c:57005 | Zbl 0412.57007
[JS] W. JACO, P. SHALEM, Seifert fibred spaces in 3-manifolds, Mem. Amer. Math. Soc., 220 (1980). Zbl 0415.57005
[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.
[Ma] W. MAGNUS, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann., 111 (1935). Zbl 0011.15201 | JFM 61.0102.02
[Ma1] A. MAL'CEV, On isomorphic matrix representations of infinite groups, Math. Sb., 8 (50) (1940), 405-422. MR 2,216d | Zbl 0025.00804
[Sa] T. SAKAI, Geodesic knots in hyperbolic 3-manifold, Kobe J. Math., 8 (1991), 81-87. MR 93a:57011 | Zbl 0749.57003
[Sc1] P. SCOTT, The geometry of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. MR 84m:57009 | Zbl 0561.57001
[Sc2] P. SCOTT, There are no fake Seifert fibred spaces with infinite π1, Ann. of Math., (2) 117 (1983), 35-70. MR 84c:57008 | Zbl 0516.57006
[St] J. STALLINGS, Homology and Central Series of Groups, J. Algebra, 2 (1965), 170-181. MR 31 #232 | Zbl 0135.05201
[Th1] W. THURSTON, Geometry and topology of 3-manifolds, Lecture Notes, Princeton University, Princeton NJ, 1978-1979.
[Th2] W. THURSTON, Three dimensionnal manifolds, Kleinian groups, and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982) 357-381. Article | Zbl 0496.57005
[Wa1] F. WALDHAUSEN, On irreducible 3-manifolds which are sufficiently large, Ann. of Math., (2) 87 (1968), 56-88. MR 36 #7146 | Zbl 0157.30603
[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. Zbl 0202.54702
[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. MR 52 #15489 | Zbl 0309.55006