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 | References | Similar Articles | Additional Information

Abstract: Let 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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Additional Information

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