Limits of relatively hyperbolic groups and Lyndon's completions

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 $G$.

20 pages, April 28, corrected typos, recompiled to make 20 pages, Accepted to the Journal of the European Math Society

References in corpus (1)

• Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Cited by in corpus (1)

• Primitivity of group rings of non-elementary torsion-free hyperbolic groups