Free products of units in algebras - I. Quaternion algebras

Let A be a quaternion algebra over a commutative unital ring. We find suffi cient conditions for pairs of units of A to generate a free group. Using th e well-known isomorphism between SO(3, R) and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Sp ecialization techniques allow us to find similar free subgroups in skew pol ynomial rings. A consequence is the following: let kG be the group algebra of a residually (torsionfree nilpotent) group G over a field k whose charac teristic is not 2. If x and y are any pair of noncommuting elements of G, a nd c, d is an element of k* then 1 + cx and 1 + dy generate a free subgroup of the Malcev-Neumann field of fractions of kG. (C) 1999 Academic Press.