On free subgroups in division rings

Abstract

Let K K be a field, let σ \sigma be an automorphism, and let δ \delta be a σ \sigma -derivation of K K . We show that the multiplicative group of nonzero elements of the division ring D = K ( x ; σ , δ ) D=K(x;\sigma ,\delta ) contains a free noncyclic subgroup unless D D is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free nonabelian solvable-by-finite groups always contain free noncyclic subgroups.