On Fully Residually- Groups

Abstract

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-𝒞 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 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 ∈ ℛ be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem 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.