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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. 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Goncalves, Javier Sanchez, Vitor O. Ferreira","submitted_at":"2018-05-22T17:40:17Z","abstract_excerpt":"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. 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