We describe finitely generated groups universally equivalent (with constants from in the language) to a given torsion-free relatively hyperbolic group with free abelian parabolics. It turns out that, as in the free group case, the group embeds into the Lyndon's completion of the group , or, equivalently, embeds into a group obtained from by finitely many extensions of centralizers. Conversely, every subgroup of containing is universally equivalent to . Since finitely generated groups universally equivalent to are precisely the finitely generated groups discriminated by , the result above gives a description of finitely generated groups discriminated by . Moreover, these groups are exactly the coordinate groups of irreducible algebraic sets over .
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. 659–680DOI 10.4171/JEMS/314