Schreier theorem on groups which split over free abelian groups

Moon, Myoungho

doi:10.1090/S0002-9939-99-05306-X

Schreier theorem on groups which split over free abelian groups
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by Myoungho Moon PDF

Proc. Amer. Math. Soc. 128 (2000), 1885-1892 Request permission

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

Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 167618, DOI 10.2307/2373173
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
J. Morgan, Uniformization theorem for 3-manifolds, Smith Conjecture edited by J. Morgan and H. Bass, Academic Press (1984), 37-125.
O. Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1928).
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
Heiner Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics, vol. 875, Springer-Verlag, Berlin, 1981. MR 643627, DOI 10.1007/BFb0090465