On the finitely generated subgroups of an amalgamated product of two groups

Burns, R. G.

doi:10.1090/S0002-9947-1972-0372043-5

On the finitely generated subgroups of an amalgamated product of two groups
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by R. G. Burns

Trans. Amer. Math. Soc. 169 (1972), 293-306

DOI: https://doi.org/10.1090/S0002-9947-1972-0372043-5

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Abstract:

Sufficient conditions are found for the free product $G$ of two groups $A$ and $B$ with an amalgamated subgroup $U$ to have the properties (1) that the intersection of each pair of finitely generated subgroups of $G$ is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of $G$ has finite index in $G$. The known results that Fuchsian groups and free products (under the obvious conditions on the factors) have properties (1) and (2) follow as instances of the main result.

References

Benjamin Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673–679. MR 199247, DOI 10.1112/jlms/s1-41.1.673
Benjamin Baumslag, Generalized free products whose two-generator subgroups are free, J. London Math. Soc. 43 (1968), 601–606. MR 233896, DOI 10.1112/jlms/s1-43.1.601
Leon Greenberg, Discrete groups of motions, Canadian J. Math. 12 (1960), 415–426. MR 115130, DOI 10.4153/CJM-1960-036-8
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
A. Howard M. Hoare, Abraham Karrass, and Donald Solitar, Subgroups of finite index of Fuchsian groups, Math. Z. 120 (1971), 289–298. MR 285619, DOI 10.1007/BF01109993
A. G. Howson, On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), 428–434. MR 65557, DOI 10.1112/jlms/s1-29.4.428
Abraham Karrass and Donald Solitar, On finitely generated subgroups of a free product, Math. Z. 108 (1969), 285–287. MR 245654, DOI 10.1007/BF01112534
A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255. MR 260879, DOI 10.1090/S0002-9947-1970-0260879-9
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
D. I. Moldavanskiĭ, The intersection of finitely generated subgroups, Sibirsk. Mat. Ž. 9 (1968), 1422–1426 (Russian). MR 0237619