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

Karrass, A.; Solitar, D.

doi:10.1090/S0002-9939-1973-0320152-5

On finitely generated subgroups which are of finite index in generalized free products
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by A. Karrass and D. Solitar

Proc. Amer. Math. Soc. 37 (1973), 22-28

DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5

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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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Über die Permutationsgruppe der natürlichen Zahlenfolge

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