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

Abstract

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.