= Intersection and join of benign subgroups
{title2=$H\cap K,\quad\langle H,K\rangle$}
For <benign subgroups> $H,K\le G$, amalgamate their finite-presentation witnesses over the finitely generated $G$, then adjoin independent <stable letters> $t,s$ centralizing their finite-generation witnesses. <Britton's lemma> identifies the subgroup generated by $G,t,s$ with the extension centralizing $H,K$. In it, $G\cap G^{ts}=H\cap K$ and $G\cap\langle G^t,G^s\rangle=\langle H,K\rangle$. Both conjugate witnesses are finitely generated, proving benignity. For the second identity, the subgroup generated by $G,G^t,G^s$ is the iterated <amalgamated free product> $G*_H G^t *_K G^s$; its normal forms show that adjoining each outer copy intersects the central copy only in the subgroup already generated by $H,K$.
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