Conjecture on solvable groups defined by redundant cyclic presentations

For every word $U\in F(x_0,x_1)$ , let $G_U=\langle x_0,x_1\mid U(x_0,x_1)U(x_1,x_0)^{-1}\rangle$ . If $G_U$ is solvable and does not contain a free subgroup of rank $2$ , then $G_U$ is isomorphic to the Baumslag--Solitar group $BS(1,m)=\langle a,t\mid tat^{-1}=a^m\rangle$ for some $m\in\{-1,0,1\}$ ; equivalently, the only exceptional parameters arising in this family are $m=-1,0,1$ .

References

Primary source

arXiv

Additional references

Solvable groups defined by redundant cyclic presentations — arXiv

group-theory

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 preprint claims to disprove the conjecture and give a complete classification, but the result has not been independently verified.

The conjecture concerns solvable groups arising from redundant cyclic presentations and restricts the exceptional parameter to $m\in\{-1,0,1\}$ .

September 8, 2026 claimed classification

An unrefereed arXiv preprint claims that the restriction $m\in\{-1,0,1\}$ is false and classifies all exceptional groups in the family, replacing the conjectural list with a complete classification.

Current status (as of September 2026): A complete classification is claimed in an unrefereed preprint, but its correctness and the resulting resolution of the conjecture remain unverified.

Sources

No solutions have been posted yet.