Free Products of Units in Algebras I. Quaternion Algebras

LetAbe a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units ofAto generate a free group. Using the well-known isomorphism betweenSO(3, ) and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A consequence is the following: letkGbe the group algebra of a residually (torsion-free nilpotent) groupGover a fieldkwhose characteristic is not 2. Ifxandyare any pair of noncommuting elements ofG, andc, d k* then 1 + cxand 1 + dygenerate a free subgroup of the Malcev–Neumann field of fractions ofkG.