Limits of relatively hyperbolic groups and Lyndon's completions

TL;DR

This paper characterizes finitely generated groups universally equivalent to a torsion-free relatively hyperbolic group with free abelian parabolics, showing they embed into Lyndon's completion and describing their structure via extensions of centralizers.

Contribution

It extends the understanding of universal equivalence and embeddings of groups in Lyndon's completion for relatively hyperbolic groups with free abelian parabolics.

Findings

Finitely generated groups universally equivalent to G embed into G^{Z[t]}

Subgroups of G^{Z[t]} containing G are universally equivalent to G

Provides a description of finitely generated groups discriminated by G

Abstract

In this paper we describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon's completion $G^{\mathbb{Z}[t]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}[t]}$ containing $G$ is universally equivalent to $G$. Since finitely generated groups universally equivalent to $G$ are precisely the finitely generated groups discriminated by $G$ the result above gives a description of finitely generated groups discriminated by $G$.