Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

  • Vitor O. FerreiraEmail author
  • Jairo Z. Gonçalves
  • Javier Sánchez


Let k be a field of characteristic different from 2 and let G be a nonabelian residually torsion-free nilpotent group. It is known that G is an orderable group. Let k(G) denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of G over k. If ∗ is an involution on G, then it extends to a unique k-involution on k(G). We show that k(G) contains pairs of symmetric elements with respect to ∗ which generate a free group inside the multiplicative group of k(G). Free unitary pairs also exist if G is torsion-free nilpotent. Finally, we consider the general case of a division ring D, with a k-involution ∗, containing a normal subgroup N in its multiplicative group, such that \(G \subseteq N\), with G a nilpotent-by-finite torsion-free subgroup that is not abelian-by-finite, satisfying G = G and N = N. We prove that N contains a free symmetric pair.


Infinite dimensional division rings Division rings with involution (Residually) torsion-free nilpotent groups 


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The authors were partially supported by grant #2015/09162-9, São Paulo Research Foundation (FAPESP) - Brazil; the second author was partially supported by CNPq - Brazil (grant 301205/2015-9); the third author was partially supported by CNPq - Brazil (grant 307638/2015-4).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of São PauloSão PauloBrazil

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