On the universal theory of torsion and lacunary hyperbolic groups

TL;DR

This paper explores the universal theories of torsion, hyperbolic, and lacunary hyperbolic groups, revealing their relationships, decidability issues, and constructing examples with undecidable theories, highlighting torsion groups' significance.

Contribution

It establishes the strong containment of torsion groups' universal theory within finite groups' theory and constructs lacunary hyperbolic groups with undecidable universal theories.

Findings

Universal theory of torsion groups is contained in that of finite groups.

Universal theories of certain torsion groups are undecidable.

Universal theory of hyperbolic groups is undecidable, enabling construction of lacunary hyperbolic groups with undecidable theories.

Abstract

We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.