AI 中文总结
该研究证明含Baumslag-Solitar子群BS(m,n)（mn≠0）的群，其带群作用的域无模型伴侣，解决了Beyarslan和Kowalski的猜想，还确定ℚ²等群的对应域无模型伴侣，改进了Hrushovski的相关方法。
AI 中文摘要
我们证明：若群G包含Baumslag-Solitar群BS(m,n)且mn≠0，则带G作用的域的理论不存在模型伴侣。特别地，每个包含ℤ²≅BS(1,1)的群，其带群作用的域都不存在模型伴侣，这解决了Beyarslan和Kowalski的一个猜想。因此，ℚ²、Thompson群F、T、V以及Higman群的带群作用的域均不存在模型伴侣。我们的论证改编了Hrushovski针对两个交换自同构的方法，关键修改是分圆条件θ_q，该条件无需指定对本原单位根的非平凡作用，从而使Hrushovski的方法在从子群过渡到 ambient群后可适用。
英文摘要
We prove that if a group $G$ contains a Baumslag-Solitar group $\mathrm{BS}(m,n)$, with $mn\neq0$, then the theory of fields with a $G$-action has no model companion. In particular, every group containing $\mathbb{Z}^2\cong\mathrm{BS}(1,1)$ has no model companion for its field actions, resolving a conjecture of Beyarslan and Kowalski. Consequently, there are no model companions for field actions of $\mathbb{Q}^2$, Thompson's groups $F,T,V$, or Higman's group. Our argument adapts Hrushovski's method for two commuting automorphisms. The key modification is the cyclotomic condition $θ_q$, which removes the need to prescribe a non-trivial action on a root of unity and thereby makes Hrushovski's method applicable after passing from a subgroup to an ambient group.
