On the free product of two groups with an amalgamated subgroup of finite index in each factor

Karrass, A.; Solitar, D.

doi:10.1090/S0002-9939-1970-0263928-2

On the free product of two groups with an amalgamated subgroup of finite index in each factor
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by A. Karrass and D. Solitar

Proc. Amer. Math. Soc. 26 (1970), 28-32

DOI: https://doi.org/10.1090/S0002-9939-1970-0263928-2

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Abstract:

Let $G = (A \ast B;U)$ where $U$ is finitely generated and of finite index $\ne 1$ in both $A$ and $B$. We prove that $G$ is a finite extension of a free group iff $A$ and $B$ are both finite. In particular, this answers in the negative a question of W. Magnus as to whether or not $G$ can be free. Analogous results are obtained for tree products and HNN groups.

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