📝 SummarXiv

Abstract

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups. We apply this to construct simple torsion-free Tarski monsters (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) with the same positive theory as the free group, answering several questions of Casals-Ruiz, Garreta, and de la Nuez Gonz\'alez. Moreover, these groups may be chosen so that every non-trivial conjugacy-invariant norm is stably bounded. In particular, for every word $w$ that is not silly in the sense of Segal, the $w$-length is unbounded, but the stable $w$-length vanishes. 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.