On free subgroups in division rings

Abstract

Let $K$ be a field and let $\ddot{I}ƒ$ be an automorphism and let $\hat{I}´$ be a $\ddot{I}ƒ$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;\ddot{I}ƒ,\hat{I}´)$ contains a free non-cyclic subgroup unless $D$ 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 non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.

nine pages