  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.
