On countable elementary free groups

TL;DR

This paper proves that countable groups elementarily equivalent to non-abelian free groups with cyclic abelian subgroups are unions of hyperbolic tower groups, revealing structural properties of such groups.

Contribution

It establishes a structural characterization of certain countable groups elementarily equivalent to free groups using hyperbolic towers.

Findings

Countable groups with specific elementary equivalence properties are unions of hyperbolic towers.

All abelian subgroups of these groups are cyclic.

Provides a new link between model theory and geometric group theory.

Abstract

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).