Subgroup separability of the figure 8 knot group

Topology

Volumn 45, Issue 3, 2006, Pages 421-463

  Wise, Daniel T ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Graphs of groups; Knots; LERF; Subgroup separable

Indexed keywords

EID: 33645953336     PISSN: 00409383     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.top.2005.06.004     Document Type: Article

References (55)

1. I. Agol, Tameness of hyperbolic 3-manifolds, 2004.
2. Agol I., Long D.D., and Reid A.W. The Bianchi groups are separable on geometrically finite subgroups. Ann. Math. (2) 153 3 (2001) 599-621
3. Aumann R.J. Asphericity of alternating knots. Ann. Math. 64 2 (1956) 374-392
4. Baumslag G., and Solitar D. Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 (1962) 199-201
5. Bestvina M., and Feighn M. A combination theorem for negatively curved groups. J. Differential Geom. 35 1 (1992) 85-101
6. Bonahon F. Bouts des variétés hyperboliques de dimension 3. Ann. Math. (2) 124 1 (1986) 71-158
7. Bridson M.R. On the existence of flat planes in spaces of nonpositive curvature. Proc. Amer. Math. Soc. 123 1 (1995) 223-235
8. Bridson M.R., and Haefliger A. Metric Spaces of Non-positive Curvature (1999), Springer, Berlin
9. Bridson M.R., and Wise D.T. VH complexes, towers and subgroups of F × F. Math. Proc. Cambridge Philos. Soc. 126 3 (1999) 481-497
10. A.M. Brunner, R.G. Burns, D. Solitar, The subgroup separability of free products of two free groups with cyclic amalgamation, in: Contributions to Group Theory, Contemporary Mathematics, vol. 33, American Mathematical Society, Providence, RI, 1984, pp. 90-115.
11. Burns R.G., Karrass A., and Solitar D. A note on groups with separable finitely generated subgroups. Bull. Austral. Math. Soc. 36 1 (1987) 153-160
12. Canary R.D. A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35 3 (1996) 751-778
13. Culler M., and Shalen P.B. Bounded, separating, incompressible surfaces in knot manifolds. Invent. Math. 75 3 (1984) 537-545
14. Dehn M. Papers on Group Theory and Topology (1987), Springer, New York, Berlin (Translated from the German and with introductions and an appendix by John Stillwell, with an appendix by Otto Schreier.)
15. R.H. Fox. Some problems in knot theory, in: Topology of 3-Manifolds and Related Topics (Proceedings of the University of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, NJ, 1962, pp. 168-176.
16. Gersten S.M. Coherence in doubled groups. Comm. Algebra 9 18 (1981) 1893-1900
17. Gitik R. Graphs and separability properties of groups. J. Algebra 188 1 (1997) 125-143
18. Gitik R. Doubles of groups and hyperbolic LERF 3-manifolds. Ann. Math. (2) 150 3 (1999) 775-806
19. Gitik R., and Rips E. On separability properties of groups. Int. J. Algebra Comput. 5 6 (1995) 703-717
20. M. Gromov, Hyperbolic groups, in: Essays in Group Theory, Mathematical Sciences Research Institute Publications, vol. 8, Springer, New York, 1987, pp. 75-263.
21. Hall Jr. M. Coset representations in free groups. Trans. Amer. Math. Soc. 67 (1949) 421-432
22. J. Hempel, Residual finiteness for 3-manifolds, in: Combinatorial Group Theory and Topology (Alta, Utah, 1984), Princeton University Press, Princeton, NJ, 1987, pp. 379-396.
23. Hruska G.C. Nonpositively curved 2-complexes with isolated flats. Geom. Topol. 8 (2004) 205-275 (electronic)
24. Hsu T., and Wise D.T. Separating quasiconvex subgroups of right-angled Artin groups. Math. Z. 240 3 (2002) 521-548
25. W.H. Kazez, (Ed.), Geometric Topology, AMS/IP Studies in Advanced Mathematics, vol. 2, American Mathematical Society, Providence, RI, 1997.
26. Long D.D. Immersions and embeddings of totally geodesic surfaces. Bull. London Math. Soc. 19 5 (1987) 481-484
27. Long D.D. Engulfing and subgroup separability for hyperbolic groups. Trans. Amer. Math. Soc. 308 2 (1988) 849-859
28. Long D.D., and Niblo G.A. Subgroup separability and 3-manifold groups. Math. Z. 207 2 (1991) 209-215
29. Long D.D., and Reid A.W. The fundamental group of the double of the figure-eight knot exterior is GFERF. Bull. London Math. Soc. 33 4 (2001) 391-396
30. Long D.D., and Reid A.W. On subgroup separability in hyperbolic Coxeter groups. Geom. Dedicata 87 1-3 (2001) 245-260
31. Lyndon R.C., and Schupp P.E. Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 89 (1977), Springer, Berlin
32. Malc'ev A.I. On homomorphisms onto finite groups. Uchen. Zap. Ivanov Gos. Ped. Inst. 18 (1956) 49-60 (English transl. in: Amer. Math. Soc. Transl. (2) 119 (1983) 67-79
33. Menasco W. Closed incompressible surfaces in alternating knot and link complements. Topology 23 1 (1984) 37-44
34. Neuwirth L.P. Knot groups. Annals of Mathematics Studies, No. 56 (1965), Princeton University Press, Princeton, NJ
35. Niblo G.A. H.N.N. extensions of a free group by z which are subgroup separable. Proc. London Math. Soc. (3) 61 1 (1990) 18-32
36. G.A. Niblo, D.T. Wise, The engulfing property for 3-manifolds, in: The Epstein birthday Schrift, Geometric Topology, Coventry, 1998, pp. 413-418 (electronic).
37. Niblo G.A., and Wise D.T. Subgroup separability knot groups and graph manifolds. Proc. Amer. Math. Soc. 129 3 (2001) 685-693
38. J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149, Springer, New York, 1994.
39. D. Rolfsen, Knots and Links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, 1990 (Corrected reprint of the 1976 original).
40. Rubinstein J.H., and Wang S. π 1-injective surfaces in graph manifolds. Comment. Math. Helv. 73 4 (1998) 499-515
41. Scott G.P. Compact submanifolds of 3-manifolds. J. London Math. Soc. (2) 7 (1973) 246-250
42. Scott P. Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17 3 (1978) 555-565
43. Scott P. Correction to: Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17 3 (1978) 555-565 MR 58 #12996, J. London Math. Soc. (2) 32 (2) (1985) 217-220
44. P. Scott, T. Wall, Topological methods in group theory, in: Homological group theory (Proceedings of the Symposium, Durham, 1977), London Mathematical Society, Lecture Note Series, vol. 36, Cambridge University Press, Cambridge, 1979, pp. 137-203.
45. Shalen P.B. Three-manifolds and Baumslag-Solitar groups. Topology Appl. 110 1 (2001) 113-118 (Geometric Topology and Geometric Group Theory, Milwaukee, WI, 1997)
46. Stallings J.R. Topology of finite graphs. Invent. Math. 71 3 (1983) 551-565
47. Susskind P., and Swarup G.A. Limit sets of geometrically finite hyperbolic groups. Amer. J. Math. 114 2 (1992) 233-250
48. W.P. Thurston, The geometry and topology of 3-manifolds. Lecture Notes.
49. Thurston W.P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 3 (1982) 357-381
50. W.P. Thurston, Three-dimensional Geometry and Topology, vol. 1, in: S. Levy (Ed.), Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997.
51. M. Tretkoff, Covering spaces, subgroup separability, and the generalized M. Hall property, in: Combinatorial Group Theory (College Park, MD, 1988), Contemporary Mathematics, vol. 109, American Mathematical Society, Providence, RI, 1990, pp. 179-191.
52. Weinbaum C.M. The word and conjugacy problems for the knot group of any tame, prime, alternating knot. Proc. Amer. Math. Soc. 30 (1971) 22-26
53. D.T. Wise, Non-positively curved squared complexes, aperiodic tilings, and non-residually finite groups. Ph.D. Thesis, Princeton University, 1996.
54. Wise D.T. Subgroup separability of graphs of free groups with cyclic edge groups. Q. J. Math. 51 1 (2000) 107-129
55. Wise D.T. The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149 3 (2002) 579-617