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Subsets of free groups contained in $2$-generated subgroups

$\DeclareMathOperator\rank{rank}$Let $F$ be a non-cyclic free group. For which finitely generated subgroups $H< F$ such that $H$ is not of finite index in a free factor of $F$ does there exist a two- …
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What can be said about a group from its presentation?

Quite a lot, I believe (although `a lot' is subjective). A neat example of a property which can be read immediately off of a (finite) presentation $\langle X; R \rangle$ is the Deficiency of said pre …
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