Dice Question Streamline Icon: https://streamlinehq.com

Derived series length and maximal residually solvable quotients for one-relator groups

Prove that for every one-relator group $G = F/{w}$ there exists a word $r \in F$ with $w \in r[{r},{r}]$ such that the derived series satisfies $G^{(\omega+1)} = G^{(\omega)} = {r}_G$, thereby establishing that the length of the derived series of any one-relator group is at most $\omega$, and characterize the maximal residually solvable quotient as $F/{r}$.

Information Square Streamline Icon: https://streamlinehq.com

Background

Recent work shows rational analogues (using the rational derived series) stabilize at stage ω\omega to a normal closure generated by a suitable root rr. Extending this to the classical derived series would unify residual solvability behavior across the whole class of one-relator groups.

References

\begin{conjecture} Let $G = F/{w}$ be a one-relator group. There is a word $r\in F$ such that $w\in r[{r}, {r}]$ and

G{(\omega+1)} = G{(\omega)} = {r}_G.

In particular, the maximal residually solvable quotient of $G$ is the one-relator group $F/{r}$. \end{conjecture}

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 7.4 (Residually solvable and rationally solvable one-relator groups)