Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3
Annales de l'Institut Fourier, Volume 48 (1998) no. 2, p. 535-551
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.
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.
@article{AIF_1998__48_2_535_0,
     author = {Dubois, Jo\"el},
     title = {N\oe uds Fox-r\'esiduellement nilpotents et rigidit\'e virtuelle des vari\'et\'es hyperboliques de dimension 3},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     pages = {535-551},
     doi = {10.5802/aif.1628},
     mrnumber = {2000a:57027},
     zbl = {0899.57008},
     mrnumber = {1625594},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1998__48_2_535_0}
}
Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3. Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 535-551. doi : 10.5802/aif.1628. https://aif.centre-mersenne.org/item/AIF_1998__48_2_535_0/

[Du] J. Dubois, Nœuds 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 192475 | 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 1205445 | MR 94b:57016 | Zbl 0801.57009

[Ga2] D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom., 18 (1983), 445-503. | MR 723813 | MR 86a:57009 | Zbl 0533.57013

[GMT] D. Gabai, G.R. Meyerhoff, N. Thurston, Homotopy Hyperbolic 3-manifolds are hyperbolic, preprint. | MR 1973051 | Zbl 1052.57019

[He] J. Hempel, 3-Manifolds, Ann. of Math. Stud., vol 86, Princeton Univ. Press, Princeton NJ, 1976. | MR 415619 | MR 54 #3702 | Zbl 0345.57001

[Hi] M.W. Hirsch, Differential topology, Springer-Verlag, Berlin-Heidelberg-New York, 1976. | MR 448362 | MR 56 #6669 | Zbl 0356.57001

[HS] J. Hass, P. Scott, Homotopy equivalence and homeomorphism of 3-manifolds, Topology, 31 (1992), 493-517. | MR 1174254 | 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 565450 | 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). | JFM 61.0102.02 | Zbl 0011.15201

[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. | 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