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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Schreier theorem on groups which split
over free abelian groups


Author: Myoungho Moon
Journal: Proc. Amer. Math. Soc. 128 (2000), 1885-1892
MSC (1991): Primary 20E06, 30F40, 57M07
Published electronically: November 1, 1999
MathSciNet review: 1652240
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be either a free product with amalgamation $A *_C B$ or an HNN group $A *_C,$ where $C$ is isomorphic to a free abelian group of finite rank. Suppose that both $A$ and $B$ have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if $G$ contains a finitely generated normal subgroup $N$ which is neither contained in $C$ nor free, then the index of $N$ in $G$ is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of $3$-manifolds $M_1$ and $M_2$, the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of $M_1$ and $M_2$ has at least one nontorus boundary.


References [Enhancements On Off] (What's this?)

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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05306-X
PII: S 0002-9939(99)05306-X
Keywords: Free product with amalgamation, HNN group, graph of groups, fundamental group, hyperbolic manifolds
Received by editor(s): September 5, 1997
Received by editor(s) in revised form: August 10, 1998
Published electronically: November 1, 1999
Additional Notes: The author was partially supported by Konkuk University Research Fund and Korean Ministry of Education Research Fund, BSRI-98-1438.
Communicated by: Ralph Cohen
Article copyright: © Copyright 2000 American Mathematical Society