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

Full-text PDF

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 denote the fundamental group of the link consisting of a chain of circles. It is shown that is not subgroup separable. Furthermore, it is shown that is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain .

**[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**-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).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20E26,
20E06,
20F34,
57M05,
57M25

Retrieve articles in all journals with MSC (2000): 20E26, 20E06, 20F34, 57M05, 57M25

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