First-order theory of torsion-free Tarski monsters

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $\mathrm{\Gamma }$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\mathrm{\exists }\mathrm{\forall }\mathrm{\exists }$-elementarily embedded into $\mathrm{\Gamma }\ast \mathbf{Z}$. In particular, such $\mathrm{\Gamma }$ have the same two-quantifier theory as $\mathrm{\Gamma }\ast \mathbf{Z}$, and hence the same positive theory as a non-abelian free group. All previously known examples of groups with the same positive theory as the free group admit a non-elementary action on a hyperbolic space, while our examples cannot act on a hyperbolic space with a loxodromic element. Along the way, we solve the one-quantifier Knight conjecture for random quotients of arbitrary torsion-free, non-elementary, hyperbolic groups in the few-relator model.