Free products of units in algebras. I: Quaternion algebras
It is well known that a non-Abelian division ring \(D\) which is finite dimensional over its center contains a non-Abelian free subgroup. This implies, of course, that a typical pair of elements of \(D\) generates a free subgroup of rank two. Nevertheless, it is not easy to exhibit a free pair of elements. The authors are looking for the pairs of elements which are either free or ``nearly free'' in quaternion algebras over domains. To be more specific they call two elements semifree if they generate the free product of the corresponding cyclic subgroups in the ambient group of units. They also introduce the notion of being semifree modulo a normal subgroup. They suggest sufficient conditions, which are expressed in terms of valuations, for a pair of elements to be either free or semifree modulo the center. They apply these results to exhibit free subgroups of the group \(\text{SO}(3,\mathbb{R})\) of real orthogonal matrices reproving and generalizing previously known results of this kind. They also exhibit free pairs in the Malcev-Neumann skewfields generated by residually torsion-free nilpotent groups and give some further applications.
scientific article; zbMATH DE number 1275265
scientific article