Classes of Groups Generalizing a Theorem of Benjamin Baumslag

TL;DR

This paper extends Baumslag's theorem to broader classes of groups, showing that fully residually $ ext{X}$ groups are equivalent to residually $ ext{X}$ and CT, including free products, hyperbolic groups, and certain one-relator groups.

Contribution

It generalizes Baumslag's theorem to new classes of groups, demonstrating the equivalence of residual properties and CT in these classes and their closure under specific constructions.

Findings

Includes free products of cyclic groups excluding the infinite dihedral group

Torsion-free hyperbolic groups satisfy the property

One-relator groups with only odd torsion also satisfy the property

Abstract

In [BB] Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman [GS] and Remeslennikov [Re] showed that this is also equivalent to being universally free, that is, having the same universal theory as the class of nonabelian free groups. This result is one of the cornerstones of the proof of the Tarksi problems. In this paper we extend the class of groups for which Benjamin Baumslag's theorem is true, that is we consider classes of groups $\X$ for which being fully residually $\X$ is equivalent to being residually $\X$ and commutative transitive. We show that the classes of groups for which this is true is quite extensive and includes free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done in [KhM]), and one-relator groups with only odd torsion. Further, the class of groups having this property is closed under certain amalgam constructions, including free products and free products with malnormal amalgamated subgroups. We also consider extensions of these classes to classes where the equivalence with universally $\X$ groups is maintained.