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 $\ast$ 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 $\ast$ 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 $\ast$, 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.