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 in 3 to this class. We also discuss two definitions of (fully) residually-L groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii 18 and Osin 22, we prove the equivalence of the two definitions for L = R. This is a generalization of the similar result obtained by Ol'shanskii for L being the class of torsion-free hyperbolic groups. Let Gamma is an element of R be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in 14 that a finitely generated fully residually-Gamma group G embeds into an iterated extension of centralizers of Gamma. We deduce from their theorem that every finitely generated fully residually-Gamma 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.
展开▼
