Magnus conjecture on isomorphisms of one‑relator groups

Determine whether two one‑relator groups G1 = F_A / << r1 >> and G2 = F_A / << r2 >> are isomorphic if and only if the relators r1 and r2 lie in the same Aut(F_A)‑orbit; equivalently, prove or refute that every isomorphism between one‑relator quotients of the same free group is induced by a free‑group automorphism mapping one defining relator to the other up to inversion and conjugacy.

Background

This is the classical isomorphism problem formulation due to Wilhelm Magnus. It asserts that isomorphisms between one‑relator quotients of a fixed free group arise precisely from automorphisms of the free group acting on relators. A positive resolution would immediately yield a decision procedure for the one‑relator isomorphism problem via Whitehead’s algorithm for automorphic equivalence in free groups.

The authors recount that later counterexamples show the conjecture is false in full generality (e.g., torus knot groups and certain Baumslag–Gersten families), but the statement remains a foundational open formulation historically and motivates many partial versions and restrictions.