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

Abstract

Let $G=F/⟨⟨r⟩⟩$ 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 $$G_m=⟨a,t|[t,a[a,t{]}^{-m}]⟩,$$ 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.

7 pages; All comments are welcome!