AI 中文总结
本文构造了一族单关系群，其关系为交换子，每个都有Baumslitar子群作为retract，从而证明了这些群不满足霍菲性、残余有限性和自动性。
AI 中文摘要
设$G=F/\langle\langle r\rangle\rangle$为一个单关系群，其中关系$r\in [F,F]$或$r=[u,v] ~(u,v\in F)$，其中$F$为有限生成的自由群。Baumslag问是否$G$是霍菲群、残余有限群或自动群。在$r\in[F,F]$的情况下，残余有限性和自动性的问题已由Olshanskii的结果给出否定答案。在本文中，我们构造了一族单关系群$$G_m=\left\langle a,t\ \middle|\ [t,a[a,t]^{-m}]\right\rangle,$$其关系都是交换子，每个都有Baumslitar子群作为 retract。这些群在两种情况下都给出了这三个问题的否定答案。
英文摘要
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.
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