Geodesic
Limits of relatively hyperbolic groups and Lyndon’s completions
Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 659-680

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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 GZ[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 GZ[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.
DOI : 10.4171/jems/314
Classification : 20-XX, 00-XX 
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     title = {Limits of relatively hyperbolic groups and {Lyndon{\textquoteright}s} completions},
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Olga Kharlampovich; Alexei Myasnikov. Limits of relatively hyperbolic groups and Lyndon’s completions. Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 659-680. doi: 10.4171/jems/314

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