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

Abstract

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.