# Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich; Alexei Myasnikov

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 3, page 659-680
- ISSN: 1435-9855

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topKharlampovich, 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 -

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