2025/08/28 by Rémi Coulon, Coulon, Rémi, Fábio Augusto Facio +3 · 1 citation
Physics and Astronomy · #Cosmology and Gravitation Theories #Black Holes and Theoretical Physics #Astro and Planetary Science
paper · pdf · doi:10.48550/arxiv.2508.21244
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 ∃ ∀ ∃-elementarily embedded into Γ∗ Z. In particular, such Γ have the same two-quantifier theory as Γ∗ 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.