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On Certain Finitely Generated Subgroups of Groups Which Split

Published online by Cambridge University Press:  20 November 2018

Myoungho Moon
Myoungho Moon*
Affiliation:
Department of Mathematics Education, Konkuk University, Seoul 143-701, Korea, e-mail: mhmoon@kkucc.konkuk.ac.kr
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Abstract

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Define a group $G$ to be in the class $S$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that ${{g}^{n\,}}\in \,K$ for all $g\,\in \,G,\,K$ has finite index in $G$. We show that a free product with amalgamation $A{{*}_{_{C}}}B$ and an $\text{HNN}$ group $A{{*}_{C}}$ belong to $S$, if $C$ is in $S$ and every subgroup of $C$ is finitely generated.

Keywords

20E0620E0857M07free product with amalgamationHNN groupgraph of groupsfundamental group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 1 , 01 March 2003 , pp. 122 - 129
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Burns, R. G., On the finitely generated subgroups of amalgamated product of two groups. Trans. Amer. Math. Soc. 169 (1972), 293–306.Google Scholar
[2] Canary, R., A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35 (1996), 751–778.Google Scholar
[3] Cohen, D. E., Subgroups of HNN groups. J. Austral. Math. Soc. 17 (1974), 394–405.Google Scholar
[4] Cohen, D. E., Combinatorial group theory: a topological approach. London Math. Society Student Texts 14, Cambridge Univ. Press, 1989.Google Scholar
[5] Griffiths, H. B., The fundamental group of a surface, and a theorem of Schreier. Acta Math. 110 (1963), 1–17.Google Scholar
[6] Hempel, J., 3-manifolds. Ann. of Math. Stud. 86, Princeton University Press, 1976.Google Scholar
[7] Karrass, A. and Solitar, D., Note on a theorem of Schreier. Proc. Amer.Math. Soc. 8(1957), 696.Google Scholar
[8] Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory. Dover Publications, 1976.Google Scholar
[9] Schreier, O., Die untergruppen der freien gruppen. Abh. Math. Sem. Univ. Hamburg 5 (1928), 161–183.Google Scholar
[10] Scott, P., The geometry of 3-manifolds. Bull. LondonMath. Soc. 15 (1983), 401–487.Google Scholar
[11] Scott, P. and Wall, T., Topological methods in group theory. Homological group theory, London Math. Soc. Lecture Notes 36, Cambridge Univ. Press 1979, 137–203.Google Scholar
[12] Serre, J.-P., Trees. Springer-Verlag, 1980.Google Scholar