# Limits of relatively hyperbolic groups and Lyndon’s completions

### Alexei Myasnikov

McGill University, Montreal, Canada### Olga Kharlampovich

McGill University, Montreal, Canada

## Abstract

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 $G$. Moreover, these groups are exactly the coordinate groups of irreducible algebraic sets over $G$.

## Cite this article

Alexei Myasnikov, Olga Kharlampovich, Limits of relatively hyperbolic groups and Lyndon’s completions. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 659–680

DOI 10.4171/JEMS/314