HNN bases and high-dimensional knots

Silver, Daniel

doi:10.1090/S0002-9939-96-03520-4

HNN bases and high-dimensional knots
HTML articles powered by AMS MathViewer

by Daniel S. Silver PDF

Proc. Amer. Math. Soc. 124 (1996), 1247-1252 Request permission

Abstract:

There exists a $3$-knot group having HNN bases of two types: bases that are arbitrarily large finitely presented and bases that are arbitrarily large finitely generated but not finitely presented. Any $n$-knot with such a group has a Seifert manifold that can be converted to a minimal one by a finite sequence of ambient $0$- and $1$-surgeries, but cannot be converted by $1$-surgeries alone.

References

Gilbert Baumslag, On abelian hopfian groups. I, Math. Z. 78 (1962), 53–54. MR 142633, DOI 10.1007/BF01195151
Gilbert Baumslag and Peter B. Shalen, Amalgamated products and finitely presented groups, Comment. Math. Helv. 65 (1990), no. 2, 243–254. MR 1057242, DOI 10.1007/BF02566605
Robert Bieri and Ralph Strebel, Almost finitely presented soluble groups, Comment. Math. Helv. 53 (1978), no. 2, 258–278. MR 498863, DOI 10.1007/BF02566077
I. M. Chiswell, Exact sequences associated with a graph of groups, J. Pure Appl. Algebra 8 (1976), no. 1, 63–74. MR 399296, DOI 10.1016/0022-4049(76)90023-2
C. McA. Gordon, Homology of groups of surfaces in the $4$-sphere, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 113–117. MR 591977, DOI 10.1017/S0305004100057996
Jonathan A. Hillman, High dimensional knot groups which are not two-knot groups, Bull. Austral. Math. Soc. 16 (1977), no. 3, 449–462. MR 645406, DOI 10.1017/S0004972700023510
Michel A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105–119. MR 0178475
J. Levine, Unknotting spheres in codimension two, Topology 4 (1965), 9–16. MR 179803, DOI 10.1016/0040-9383(65)90045-5
T. Maeda, Knotted surfaces in the $4$-sphere with no minimal Seifert manifolds, preprint.
D. I. Moldavanskiĭ, Certain subgroups of groups with one defining relation, Sibirsk. Mat. Ž. 8 (1967), 1370–1384 (Russian). MR 0220810
B.H. Neumann, Some remarks on infinite groups, J. London Math. Soc. 12 (1937), 4–11.
E.S. Rapaport, Knot-like groups, Annals of Math. Studies, vol. 84, Princeton Univ. Press, Princeton, 1975, pp. 119–133.
Daniel S. Silver, Examples of $3$-knots with no minimal Seifert manifolds, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 417–420. MR 1120476, DOI 10.1017/S0305004100070481
Guang Fu Cao and Cheng Zu Zou, Stable convergence of operator-valued functions, Acta Math. Sinica 35 (1992), no. 6, 853–858 (Chinese, with Chinese summary). MR 1219389
D. S. Silver, On knot-like groups and ribbon concordance, J. Pure Appl. Algebra 82 (1992), no. 1, 99–105. MR 1181096, DOI 10.1016/0022-4049(92)90013-6
John Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541–543. MR 158917, DOI 10.2307/2373106