TY - JOUR
T1 - Limits of relatively hyperbolic groups and Lyndon's completions
AU - Kharlampovich, Olga
AU - Myasnikov, Alexei
PY - 2012
Y1 - 2012
N2 - 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 Lyndon's completion G 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 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.
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 Lyndon's completion G 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 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.
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U2 - 10.4171/JEMS/314
DO - 10.4171/JEMS/314
M3 - Article
AN - SCOPUS:84860627425
SN - 1435-9855
VL - 14
SP - 659
EP - 680
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
IS - 3
ER -