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Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
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 c
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