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

2026/07/23 by Ke Wang, Qiang Zhang
#math.GR

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Abstract

Let G=F/⟨⟨ r⟩⟩ be a one-relator group with the relator r∈ [F,F] or r=[u,v] ~(u,v∈ F), where F is a finitely generated free group. Baumslag asked whether G is Hopfian, residually finite or automatic. In the case of r∈[F,F], a negative answer to the residual finiteness and automaticity has already been obtained by a result of Olshanskii. In this note, we construct a family of one-relator groups G_m=⟨ a,t \middle| [t,a[a,t]^-m]⟩, whose relators are commutators, each of which has a Baumslag-Solitar subgroup as a retract. These groups provide negative answers to these three questions in both cases.

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