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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).

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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)