Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

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?)

  • 1. Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 0167618
  • 2. Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
  • 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. Peter Scott and Terry Wall, Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 137–203. MR 564422
  • 6. Heiner Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics, vol. 875, Springer-Verlag, Berlin, 1981. MR 643627

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20E06, 30F40, 57M07

Retrieve articles in all journals with MSC (1991): 20E06, 30F40, 57M07

Additional Information

Myoungho Moon
Affiliation: Department of Mathematics Education, Konkuk University, Seoul 143-701, Korea

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