On fully residually-R groups

Original · EN

We consider the class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Benjamin Baumslag to this class. We also discuss two definitions of (fully) residually-C groups and prove the equivalence of the two definitions for C=R. This is a generalization of the similar result obtained by Ol'shanskii for C being the class of torsion-free hyperbolic groups. Let Γ be non-abelian and non-elementary. We prove that every finitely generated fully residually-Γ group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.