Limits of relatively hyperbolic groups and Lyndon’s completions

Kharlampovich, Olga, and Myasnikov, Alexei. "Limits of relatively hyperbolic groups and Lyndon’s completions." Journal of the European Mathematical Society 014.3 (2012): 659-680. <http://eudml.org/doc/277695>.

@article{Kharlampovich2012,
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 $G$.},
author = {Kharlampovich, Olga, Myasnikov, Alexei},
journal = {Journal of the European Mathematical Society},
keywords = {finitely generated groups; universal equivalences; relatively hyperbolic groups; Lyndon completions; extensions of centralizers; coordinate groups of irreducible algebraic sets; finitely generated groups; universal equivalences; relatively hyperbolic groups; Lyndon completions; extensions of centralizers; coordinate groups of irreducible algebraic sets},
language = {eng},
number = {3},
pages = {659-680},
publisher = {European Mathematical Society Publishing House},
title = {Limits of relatively hyperbolic groups and Lyndon’s completions},
url = {http://eudml.org/doc/277695},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Kharlampovich, Olga
AU - Myasnikov, Alexei
TI - Limits of relatively hyperbolic groups and Lyndon’s completions
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 3
SP - 659
EP - 680
AB - 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 $G$.
LA - eng
KW - finitely generated groups; universal equivalences; relatively hyperbolic groups; Lyndon completions; extensions of centralizers; coordinate groups of irreducible algebraic sets; finitely generated groups; universal equivalences; relatively hyperbolic groups; Lyndon completions; extensions of centralizers; coordinate groups of irreducible algebraic sets
UR - http://eudml.org/doc/277695
ER -