Decidability of conjugacy and isomorphism problems for one‑relator groups

Determine the algorithmic decidability of the conjugacy problem and the isomorphism problem for arbitrary one‑relator groups.

Background

The authors emphasize that, despite Magnus’s solution of the word problem, the conjugacy and isomorphism problems in the full class of one‑relator groups have resisted resolution. Many subclasses are known (e.g., hyperbolic one‑relator groups), but no general decision procedures exist.

They flag these as enduring open problems central to the theory.