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

Abstract

Let $G=F/\langle\langle r\rangle\rangle$ be a one-relator group with the relator $r\in [F,F]$ or $r=[u,v] ~(u,v\in F)$, where $F$ is a finitely generated free group. Baumslag asked whether $G$ is Hopfian, residually finite or automatic. In the case of $r\in[F,F]$, 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 $$G_m=\left\langle a,t\ \middle|\ [t,a[a,t]^{-m}]\right\rangle,$$ 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.