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A note on groups with separable finitely generated subgroups

Published online by Cambridge University Press:  17 April 2009

R. G. Burns ,
A. Karrass  and
D. Solitar
R. G. Burns
Affiliation:
Department of Mathematics, York University, North York, Ontario, M3J 1P3Canada.
A. Karrass
Affiliation:
Department of Mathematics, York University, North York, Ontario, M3J 1P3Canada.
D. Solitar
Affiliation:
Department of Mathematics, York University, North York, Ontario, M3J 1P3Canada.
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Abstract

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An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 1 , August 1987 , pp. 153 - 160
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Allenby, R.B.J.T. and Gregorac, R.J., “On locally extended residually finite groups”, Conference on Group Theory, Lecture Notes in Mathematics, 319 (Springer Verlag, Berlin-Heidelberg-New York 1973) 9–17.Google Scholar
[2]Bieri, R., Homological Dimension of Discrete Groups (Second Edition) (Queen Mary College Mathematics Notes, University of London, 1981).Google Scholar
[3]Brunner, A.M., Burns, R.G. and Solitar, D., “The subgroup separability of free products of two free groups with cyclic amalgamation”, Contemp. Math. 33 (1984) 90–115.CrossRefGoogle Scholar
[4]Karrass, A., Pietrowski, A. and Solitar, D., “An improved subgroup theorem for HNN groups with some applications”, Canad. J. Math. 26 (1974), 214–224.CrossRefGoogle Scholar
[5]Karrass, A. and Solitar, D., “The subgroups of a free product of two groups with an amalgamated subgroup”, Trans. Amer. Math. Soc. 149 (1970), 227–255.CrossRefGoogle Scholar
[6]Pietrowski, A., “The isomorphism problem for one-relator groups having non-trivial centre”, Math. Z. 136 (1974), 95–106.CrossRefGoogle Scholar
[7]Papakyriakopoulos, C.D., “On Dehn's lemma and the asphericity of knots”, Ann. of Math. 66 (1957), 1–26.CrossRefGoogle Scholar
[8]Scott, G.P., “Finitely generated 3-manifold groups are finitely presented”, J. London Math. Soc. 6 (1973), 437–440.CrossRefGoogle Scholar