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Questions about the branch of algebra that deals with groups.

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Examples of a group with infinitely many ends which are not represented as a free product of...

Let $F_1$ and $F_2$-non-trivial groups. Is it correct that the number of ends of the free product $F_1\ast F_2$ is infinite? My thoughts about this: Since $e(G)=\infty$ then $G=F_1\ast F_2$, a non-t …
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