Magnus two‑relator Nielsen–move conjecture

Establish whether for free groups F_A and two pairs of relators {r1, r2} and {r1', r2'}, equality of normal closures <<r1, r2>> = <<r1', r2'>> in F_A implies that {r1, r2} can be transformed to {r1', r2'} by a finite sequence of Nielsen moves (multiplying one relator by another or its inverse, inversions, and conjugations).

Background

Magnus proposed extending his conjugacy theorem from one relator to the case of two relators by allowing Nielsen transformations on pairs. Such a result would have strong implications, including connections to the Andrews–Curtis conjecture for balanced trivial presentations.

The authors note this two‑relator analogue remains unproven and is a longstanding problem in combinatorial group theory.

References

Specifically, he conjectured that if two pairs of relators ${ r_1, r_2 }$ and ${ r_1', r_2' }$ give rise to the same normal closures in $F_A$, then the two sets are connected by a set of ``Nielsen moves'', i.e. the closure of maps of the form $r_i \mapsto r_i r_j{\pm 1}$ (with $i \neq j$), inversions, and conjugates. The conjecture remains, to the best of the authors' knowledge, open.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 3.2 (Applications of the Freiheitssatz)