First-order theory of torsion-free Tarski monsters
Rémi Coulon, Francesco Fournier-Facio, Meng-Che "Turbo" Ho
TL;DR
The paper develops a framework to tightly control the first-order theory of groups built as direct limits of torsion-free hyperbolic groups via geometric small cancellation quotients. It constructs a lacunary hyperbolic, simple, torsion-free Tarski monster $\Gamma_\infty$ that is $\exists\forall\exists$-elementarily embedded into $\Gamma_\infty\ast \mathbb{Z}$, yielding the same $\forall\exists$-theory as the free product and, in particular, a trivial positive theory. The authors prove a lifting theorem for morphisms through strengthened small cancellation quotients, develop limit-group and shortening machinery, and apply these to obtain a one-quantifier Knight conjecture for random quotients and a suite of properties for $\Gamma_\infty$ (e.g., vanishing stable commutator length, no unbounded quasimorphisms, and a trivial second bounded cohomology kernel). They further show that the two-quantifier theory can fail to detect acylindrical hyperbolicity, while preserving stronger embedding properties, thereby addressing long-standing questions about the logical profiles of exotic groups. Overall, the work provides a robust method to derive precise first-order control over complex limit groups and constructs new algebraic objects with tightly constrained logical and geometric features.
Abstract
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 $Γ$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $Γ\ast \mathbf{Z}$. In particular, such $Γ$ have the same two-quantifier theory as $Γ\ast \mathbf{Z}$, 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.