Superseded: Rosenblatt's Lemmas 4.8-4.9 without the abelian hypothesis — use Chou.fg_of_isVirtuallyPolycyclic_quotient_of_not_hasFreeSubsemigroupOfRankTwo

Let $B$ be a finitely generated group with no free subsemigroup on two generators (no pair $a, b \in B$ such that distinct positive words in $a, b$ give distinct elements), and let $A$ be a normal subgroup of $B$ such that the quotient $B/A$ is finitely presented. Then $A$ is finitely generated.

This is the “no free subsemigroup” form of Milnor's Lemmas 1 and 2: Rosenblatt's Lemma 4.8 (for each $a \in A$, $b \in B$ the conjugates $b^k a b^{-k}$ span a finitely generated subgroup) combined with Lemma 4.9 ($A$ is normally generated by finitely many elements), as Chou applies them in the proof of Theorem 3.2′ without the hypothesis that $A$ is abelian. The statement is not asserted in this form by either paper; it is the step Chou's induction needs.

Superseded. The same correction as for the exponentially bounded twin: Rosenblatt's Lemma 4.9 gives normal generation, not finite generation, and the assembly needs a normal form for the quotient that a finite presentation does not supply. The form Chou's proof uses is Chou.fg_of_isVirtuallyPolycyclic_quotient_of_not_hasFreeSubsemigroupOfRankTwo.