Subgroup separability, knot groups and graph manifolds

Niblo, Graham; Wise, Daniel

doi:10.1090/S0002-9939-00-05574-X

Subgroup separability, knot groups and graph manifolds
HTML articles powered by AMS MathViewer

by Graham A. Niblo and Daniel T. Wise PDF

Proc. Amer. Math. Soc. 129 (2001), 685-693 Request permission

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 $L$.

References

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, 1973, pp. 9–17. MR 0382450
A. M. Brunner, R. G. Burns, and D. Solitar, The subgroup separability of free products of two free groups with cyclic amalgamation, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 90–115. MR 767102, DOI 10.1090/conm/033/767102
R. G. Burns, A. Karrass, and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Austral. Math. Soc. 36 (1987), no. 1, 153–160. MR 897431, DOI 10.1017/S0004972700026393
Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI 10.1007/BF01231540
David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
D. D. Long and G. A. Niblo, Subgroup separability and $3$-manifold groups, Math. Z. 207 (1991), no. 2, 209–215. MR 1109662, DOI 10.1007/BF02571384
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Graham A. Niblo and Daniel T. Wise, The engulfing property for $3$-manifolds, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 413–418. MR 1668320, DOI 10.2140/gtm.1998.1.413
Derek J. S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1993. MR 1261639
D. Rolfson, Knots and Links, Publish or Perish, Inc., Houston TX, 1990.
J. Hyam Rubinstein and Shicheng Wang, $\pi _1$-injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998), no. 4, 499–515. MR 1639876, DOI 10.1007/s000140050066
Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
D.T. Wise, Subgroup separability of the figure 8 knot group, Preprint (1998).