On fully residually-$\mathcal{R}$ groups

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