JournalsjemsVol. 14, No. 3pp. 659–680

Limits of relatively hyperbolic groups and Lyndon’s completions

  • Alexei Myasnikov

    McGill University, Montreal, Canada
  • Olga Kharlampovich

    McGill University, Montreal, Canada
Limits of relatively hyperbolic groups and Lyndon’s completions cover
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Abstract

We describe finitely generated groups HH universally equivalent (with constants from GG in the language) to a given torsion-free relatively hyperbolic group GG with free abelian parabolics. It turns out that, as in the free group case, the group HH embeds into the Lyndon's completion GZ[t]G^{\mathbb{Z}[t]} of the group GG, or, equivalently, HH embeds into a group obtained from GG by finitely many extensions of centralizers. Conversely, every subgroup of GZ[t]G^{\mathbb{Z}[t]} containing GG is universally equivalent to GG. Since finitely generated groups universally equivalent to GG are precisely the finitely generated groups discriminated by GG, the result above gives a description of finitely generated groups discriminated by GG. Moreover, these groups are exactly the coordinate groups of irreducible algebraic sets over GG.

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