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

DOI:
https://doi.org/10.1090/S0002-9939-99-05306-X

Published electronically:
November 1, 1999

MathSciNet review:
1652240

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Abstract | References | Similar Articles | Additional Information

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

**1.**L. V. Ahlfors,*Finitely generated Kleinian groups*, Amer. J. Math. 86 (1964), 413-429. MR**29:4890****2.**B. Maskit,*Kleinian Groups*, Springer-Verlag (1987), Berlin. MR**90a:30132****3.**J. Morgan,*Uniformization theorem for 3-manifolds*, Smith Conjecture edited by J. Morgan and H. Bass, Academic Press (1984), 37-125. CMP**17:01****4.**O. Schreier,*Die Untergruppen der freien Gruppen*, Abh. Math. Sem. Univ. Hamburg 5 (1928).**5.**P. Scott and T. Wall,*Topological methods in group theory*, Homological group theory, London Math. Soc. Lecture Notes 36, Cambridge Univ. Press (1979), 137-203. MR**81m:57002****6.**H. Zieschang,*Finite groups of mapping classes of surfaces*, Lecture notes in mathematics, Springer-Verlag, Berlin (1981). MR**86g:57001**

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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:
https://doi.org/10.1090/S0002-9939-99-05306-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