# First-order theory of torsion-free Tarski monsters

## TL;DR

This paper develops methods to analyze the first-order theories of certain torsion-free hyperbolic groups, constructs new torsion-free Tarski monsters with specific logical embeddings, and solves the one-quantifier Knight conjecture for random quotients.

## Contribution

It introduces new techniques for controlling the first-order theories of direct limits of hyperbolic groups and constructs novel torsion-free Tarski monsters with unique logical properties.

## Key findings

- Constructed torsion-free Tarski monsters with specific elementary embeddings.
- Showed these monsters share the same two-quantifier theory as their free product with Z.
- Solved the one-quantifier Knight conjecture for random quotients of hyperbolic groups.

## 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 $\Gamma$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $\Gamma \ast \mathbf{Z}$. In particular, such $\Gamma$ have the same two-quantifier theory as $\Gamma \ast \mathbf{Z}$, 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.