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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On finitely generated subgroups which are of finite index in generalized free products


Authors: A. Karrass and D. Solitar
Journal: Proc. Amer. Math. Soc. 37 (1973), 22-28
MSC: Primary 20F05; Secondary 20E30
DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5
MathSciNet review: 0320152
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Abstract: Let $ G = (A \ast B;\;U)$ be the free product of A and B with the subgroup U amalgamated. Various conditions are given which imply that every finitely generated subgroup H containing a (nontrivial) normal subgroup of G has finite index in G (in such a case we say G has the f.g.c.n. property). In particular, if A is a noncyclic free group and U is cyclic, then G has the f.g.c.n. property. We use this last result to give a combinatorial proof that Fuchsian groups have the f.g.c.n. property; this was first proved by Greenberg using non-Euclidean geometry.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5
Keywords: Generalized free products, amalgamated products, small subgroups
Article copyright: © Copyright 1973 American Mathematical Society