Group properties forced by literal commutator relators

Must a one-relator group whose relator is literally a commutator be Hopfian, residually finite or automatic? These are separate properties, and a literal commutator is more restrictive than an arbitrary element of the free group’s commutator subgroup.

Why it matters and what counts as progress

Why it matters: Literal commutator relators are a narrower condition than membership in a commutator subgroup, so earlier broader counterexamples do not suffice.

What counts as progress: Prove one of the three properties under additional hypotheses or give explicit commutator-relator groups where it fails.

How the problem developed

1999 · Open problem [1]
Gilbert Baumslag Asked whether one-relator groups whose relator lies in the commutator subgroup must be Hopfian, residually finite, or automatic.
23 Jul 2026 · Counterexample family [2]
Ke Wang and Qiang Zhang Constructed commutator-relator one-relator groups answering all three questions negatively, including literal relators of the form [u,v]; earlier work had already answered residual finiteness and automaticity negatively for the broader commutator-subgroup case.

AI contribution

Commutator relators of one-relator groups do not force Hopficity, residual finiteness, or automaticity

Ke Wang, Qiang Zhang

For G_m=⟨a,t | [t,a[a,t]^(−m)]⟩, a Baumslag–Solitar group BS(m,m+1) is a retract. In particular, G_2 is non-Hopfian and non-residually finite, while G_1 is non-automatic. Literal commutator relators therefore do not force any of these properties.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull