Are 2-generated metabelian groups of exponent $n$ finite?

Question

Are 2-generated metabelian groups of exponent $n$ finite?

Ie, let $F$ be the free group on two generators, $n > 1$ an integer, and consider the characteristic quotient $F/F''F^n$.

Is this a finite group?

Answer 1

Yes, in fact more is true: Any finitely generated torsion solvable group is finite.

This is certainly true for abelian groups. For a general finitely generated torsion solvable group $G$, we induct on the derived length of $G$. Suppose the derived length of $G$ is $n$, and the statement is true for groups of derived length $\le n-1$. Then, $G'$ has derived length $n-1$, and is certainly torsion and solvable. Further, since $G/G'$ is abelian torsion finitely generated, $G/G'$ is finite, hence $G'$ is finite index inside $G$, so $G'$ is also finitely generated. Thus, by the induction hypothesis $G'$ is finite. Since $G$ is a finite extension of $G'$, thus $G$ is also finite.