Free algebras and free groups in Ore extensions and free group algebras in division rings

Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ-derivation of K  . We show that the division ring D=K(x;σ,δ)D=K(x;σ,δ) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over its center. In the case when K is finitely generated over a subfield k we then see that for σ a k-algebra automorphism of K and δ a k-linear derivation of K  , K(x;σ)K(x;σ) having a free subalgebra on two generators is equivalent to σ   having infinite order, and K(x;δ)K(x;δ) having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k   of characteristic zero and D⁎D⁎ contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k-algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k, then D contains the k-group algebra of the free group of rank two.