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Let A be a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units of A to generate a free group. Using the Ž . well-known isomorphism between SO 3, ‫ޒ‬ 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 Ž consequence is the following: let kG be the group algebra of a residually torsion-. free nilpotent group G over a field k whose characteristic is not 2. If x and y are any pair of noncommuting elements of G, and c, d g k U then 1 q cx and 1 q dy generate a free subgroup of the Malcev᎐Neumann field of fractions of kG. ᮊ 1999 Academic Press U

We investigate the problem of explicitly constructing non-cyclic free groups in finite-dimensional crossed products using valuation criteria. The results are applied to produce explicit free groups in division algebras generated by nilpotent groups, and symmetric free groups in group rings of finite groups in arbitrary characteristic. ᮊ