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 PDF

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

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.

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