  We study the existence of free subalgebras in division algebras, and prove
the following general result: if $A$ is a noetherian domain which is countably
generated over an uncountable algebraically closed field $k$ of characteristic
0, then either the quotient division algebra of $A$ contains a free algebra on
two generators, or it is left algebraic over every maximal subfield. As an
application, we prove that if $k$ is an uncountable algebraically closed field
and $A$ is a finitely generated $k$-algebra that is a domain of GK-dimension
strictly less than 3, then either $A$ satisfies a polynomial identity, or the
quotient division algebra of $A$ contains a free $k$-algebra on two generators.
