Limits of relatively hyperbolic groups and Lyndon's completions

Details Limits of relatively hyperbolic groups and Lyndon's completions Limits of relatively hyperbolic groups and Lyndon's completions

2009-04-16

arxiv.org/abs/0904.2423v4

Mathematics

Group Theory

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

Scientific paper

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 $G$.

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