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

TL;DR

This paper studies groups related to finitely generated toral relatively hyperbolic groups, proving their properties, equivalences of definitions, and embedding results, thereby extending classical theorems to broader classes of groups.

Contribution

It generalizes Baumslag's theorem to the class of finitely generated toral relatively hyperbolic groups and establishes the equivalence of two definitions of residually-$ ext{class}$ groups for this class.

Findings

Groups in $ ext{class}$ are commutative transitive.

Finitely generated fully residually-$ ext{non-abelian}$ groups embed into $ ext{class}$ groups.

Counterexample of a torsion-free fully residually hyperbolic group not embedding into $ ext{class}$.

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.