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

Abstract

Let $K$ be a field of characteristic zero, let $\sigma$ be an automorphism of $K$ and let $\delta$ be a $\sigma$-derivation of $K$. We show that the division ring $D=K(x;\sigma,\delta)$ 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 $k$ we then see that for $\sigma$ a $k$-algebra automorphism of $K$ and $\delta$ a $k$-linear derivation of $K$, $K(x;\sigma)$ having a free subalgebra on two generators is equivalent to $\sigma$ having infinite order, and $K(x;\delta)$ having a free subalgebra is equivalent to $\delta$ being nonzero. As an application, we show that if $D$ is a division ring with center $k$ of characteristic zero and $D^*$ 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.