Limits of relatively hyperbolic groups and Lyndon’s completions

In this paper 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 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 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.