Classes of Groups Generalizing a Theorem of Benjamin Baumslag

Abstract

Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman and Remeslennikov 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 Tarski problems. In this article, we provide new examples of groups for which Benjamin Baumslag's theorem is true, that is, we consider classes of groups 𝒳 for which a group is fully residually 𝒳 if and only if it is residually 𝒳 and commutative transitive.

We show that this is true for many important classes of groups, including those of free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done by Kharlamapovich and Myasnikov), and one-relator groups with only odd torsion. Furthermore, we show that many of the properties discussed here are closed under taking free products. We then consider the classes of groups 𝒳 for which Baumslag's conditions are also equivalent to being universally 𝒳.

Key Words:

• Fully residually free groups
• Limit groups
• Commutative transitivity

2010 Mathematics Subject Classification:

• 20E06
• 20E08
• 20E26
• 20F65
• 20F70

ACKNOWLEDGMENTS

The authors would like to thank the referee for the careful reading and the comments and suggestions that led to the present form of the article.