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Subgroup separability, knot groups and graph manifolds


Authors: Graham A. Niblo and Daniel T. Wise
Journal: Proc. Amer. Math. Soc. 129 (2001), 685-693
MSC (2000): Primary 20E26, 20E06, 20F34, 57M05, 57M25
DOI: https://doi.org/10.1090/S0002-9939-00-05574-X
Published electronically: April 27, 2000
MathSciNet review: 1707529
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let $L$ denote the fundamental group of the link consisting of a chain of $4$ circles. It is shown that $L$ is not subgroup separable. Furthermore, it is shown that $L$ is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain $L$.


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  • [AG] R.B.J.T. Allenby and R.J. Gregorac, On locally extended residually finite groups, Conference on Group Theory (Univ. Wisconsin-Parkside, Kenosha, Wis., 1972). Lecture Notes in Math., Vol. 319, Springer, Berlin, 9-17. MR 52:3333
  • [BBS] A.M. Brunner, R.G. Burns, and D. Solitar, The subgroup separability of free products of two free groups with cyclic amalgamation, Contemp. Math. 33 (1984), 90-115. MR 86e:20033
  • [BKS] R.G. Burns, A. Karrass, and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Aust. Math. Soc. 36 (1987), 153-160. MR 88g:20057
  • [BZ] G. Burde and H. Zieschang, Knots, Walter de Gruyter, Berlin - NY, 1985. MR 87b:57004
  • [CJ] A. Casson and D. Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), 441-456. MR 96f:57011
  • [G] D. Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992), 447-510. MR 93m:20065
  • [LN] D.D. Long and G.A. Niblo, Subgroup Separability and 3-manifold groups, Math. Z. 207 (1991), 209-215. MR 92g:20047
  • [LS] R.C. Lyndon, and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977. MR 58:28182
  • [NW] G.A. Niblo and D.T. Wise, The engulfing property for 3-manifolds, in: The Epstein birthday schrift (1998), 413-418. MR 99k:57041
  • [Rob] D.J.S. Robinson, A Course in the Theory of Groups (Graduate Texts in Math., vol 80), Springer-Verlag, 1993. MR 94m:20001
  • [Rol] D. Rolfson, Knots and Links, Publish or Perish, Inc., Houston TX, 1990.
  • [RW] J.H. Rubinstein and S. Wang, On $\pi _{1}$-injective surfaces in graph manifolds, Comm. Math. Helv. 73 (1998), 499-515. MR 99h:57039
  • [S] P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1979), 555-565. MR 58:12996; correction MR 87k:57003
  • [T] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357-382. MR 83h:57019
  • [W] D.T. Wise, Subgroup separability of the figure 8 knot group, Preprint (1998).

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Additional Information

Graham A. Niblo
Affiliation: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton, SO17 1BJ, England
Email: gan@maths.soton.ac.uk

Daniel T. Wise
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Address at time of publication: Department of Mathematics, White Hall, Cornell University, Ithaca, New York 14853
Email: daniwise@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05574-X
Received by editor(s): April 14, 1998
Received by editor(s) in revised form: May 24, 1999
Published electronically: April 27, 2000
Additional Notes: The second author was supported as an NSF Postdoctoral Fellow under grant no. DMS-9627506.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

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